FFT - 快速傅里叶变换

2022-08-10 15:32:00 阅读：216 来源： 互联网

标签：md cm 变换 border FFT color background 0px 傅里叶

<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>

.CodeMirror { height: auto; }
.CodeMirror.cm-s-inner { background: inherit; }
.CodeMirror-scroll { overflow: auto hidden; z-index: 3; }
.CodeMirror-gutter-filler, .CodeMirror-scrollbar-filler { background-color: rgb(255, 255, 255); }
.CodeMirror-gutters { border-right: 1px solid rgb(221, 221, 221); background: inherit; white-space: nowrap; }
.CodeMirror-linenumber { padding: 0px 3px 0px 5px; text-align: right; color: rgb(153, 153, 153); }
.cm-s-inner .cm-keyword { color: rgb(119, 0, 136); }
.cm-s-inner .cm-atom, .cm-s-inner.cm-atom { color: rgb(34, 17, 153); }
.cm-s-inner .cm-number { color: rgb(17, 102, 68); }
.cm-s-inner .cm-def { color: rgb(0, 0, 255); }
.cm-s-inner .cm-variable { color: rgb(0, 0, 0); }
.cm-s-inner .cm-variable-2 { color: rgb(0, 85, 170); }
.cm-s-inner .cm-variable-3 { color: rgb(0, 136, 85); }
.cm-s-inner .cm-string { color: rgb(170, 17, 17); }
.cm-s-inner .cm-property { color: rgb(0, 0, 0); }
.cm-s-inner .cm-operator { color: rgb(152, 26, 26); }
.cm-s-inner .cm-comment, .cm-s-inner.cm-comment { color: rgb(170, 85, 0); }
.cm-s-inner .cm-string-2 { color: rgb(255, 85, 0); }
.cm-s-inner .cm-meta { color: rgb(85, 85, 85); }
.cm-s-inner .cm-qualifier { color: rgb(85, 85, 85); }
.cm-s-inner .cm-builtin { color: rgb(51, 0, 170); }
.cm-s-inner .cm-bracket { color: rgb(153, 153, 119); }
.cm-s-inner .cm-tag { color: rgb(17, 119, 0); }
.cm-s-inner .cm-attribute { color: rgb(0, 0, 204); }
.cm-s-inner .cm-header, .cm-s-inner.cm-header { color: rgb(0, 0, 255); }
.cm-s-inner .cm-quote, .cm-s-inner.cm-quote { color: rgb(0, 153, 0); }
.cm-s-inner .cm-hr, .cm-s-inner.cm-hr { color: rgb(153, 153, 153); }
.cm-s-inner .cm-link, .cm-s-inner.cm-link { color: rgb(0, 0, 204); }
.cm-negative { color: rgb(221, 68, 68); }
.cm-positive { color: rgb(34, 153, 34); }
.cm-header, .cm-strong { font-weight: 700; }
.cm-del { text-decoration: line-through; }
.cm-em { font-style: italic; }
.cm-link { text-decoration: underline; }
.cm-error { color: red; }
.cm-invalidchar { color: red; }
.cm-constant { color: rgb(38, 139, 210); }
.cm-defined { color: rgb(181, 137, 0); }
div.CodeMirror span.CodeMirror-matchingbracket { color: rgb(0, 255, 0); }
div.CodeMirror span.CodeMirror-nonmatchingbracket { color: rgb(255, 34, 34); }
.cm-s-inner .CodeMirror-activeline-background { background: inherit; }
.CodeMirror { position: relative; overflow: hidden; }
.CodeMirror-scroll { height: 100%; outline: 0px; position: relative; box-sizing: content-box; background: inherit; }
.CodeMirror-sizer { position: relative; }
.CodeMirror-gutter-filler, .CodeMirror-hscrollbar, .CodeMirror-scrollbar-filler, .CodeMirror-vscrollbar { position: absolute; z-index: 6; display: none; outline: 0px; }
.CodeMirror-vscrollbar { right: 0px; top: 0px; overflow: hidden; }
.CodeMirror-hscrollbar { bottom: 0px; left: 0px; overflow: auto hidden; }
.CodeMirror-scrollbar-filler { right: 0px; bottom: 0px; }
.CodeMirror-gutter-filler { left: 0px; bottom: 0px; }
.CodeMirror-gutters { position: absolute; left: 0px; top: 0px; padding-bottom: 10px; z-index: 3; overflow-y: hidden; }
.CodeMirror-gutter { white-space: normal; height: 100%; box-sizing: content-box; padding-bottom: 30px; margin-bottom: -32px; display: inline-block; }
.CodeMirror-gutter-wrapper { position: absolute; z-index: 4; background: 0px 0px !important; border: none !important; }
.CodeMirror-gutter-background { position: absolute; top: 0px; bottom: 0px; z-index: 4; }
.CodeMirror-gutter-elt { position: absolute; cursor: default; z-index: 4; }
.CodeMirror-lines { cursor: text; }
.CodeMirror pre { border-radius: 0px; border-width: 0px; background: 0px 0px; font-family: inherit; font-size: inherit; margin: 0px; white-space: pre; overflow-wrap: normal; color: inherit; z-index: 2; position: relative; overflow: visible; }
.CodeMirror-wrap pre { overflow-wrap: break-word; white-space: pre-wrap; word-break: normal; }
.CodeMirror-code pre { border-right: 30px solid transparent; width: fit-content; }
.CodeMirror-wrap .CodeMirror-code pre { border-right: none; width: auto; }
.CodeMirror-linebackground { position: absolute; inset: 0px; z-index: 0; }
.CodeMirror-linewidget { position: relative; z-index: 2; overflow: auto; }
.CodeMirror-wrap .CodeMirror-scroll { overflow-x: hidden; }
.CodeMirror-measure { position: absolute; width: 100%; height: 0px; overflow: hidden; visibility: hidden; }
.CodeMirror-measure pre { position: static; }
.CodeMirror div.CodeMirror-cursor { position: absolute; visibility: hidden; border-right: none; width: 0px; }
.CodeMirror div.CodeMirror-cursor { visibility: hidden; }
.CodeMirror-focused div.CodeMirror-cursor { visibility: inherit; }
.cm-searching { background: rgba(255, 255, 0, 0.4); }
span.cm-underlined { text-decoration: underline; }
span.cm-strikethrough { text-decoration: line-through; }
.cm-tw-syntaxerror { color: rgb(255, 255, 255); background-color: rgb(153, 0, 0); }
.cm-tw-deleted { text-decoration: line-through; }
.cm-tw-header5 { font-weight: 700; }
.cm-tw-listitem:first-child { padding-left: 10px; }
.cm-tw-box { border-style: solid; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-color: inherit; border-top-width: 0px !important; }
.cm-tw-underline { text-decoration: underline; }
@media print {
.CodeMirror div.CodeMirror-cursor { visibility: hidden; }
}

:root {
--mermaid-theme: night;
}

[lang='mermaid'] .label {
color: #333;
}

/* CSS Document */

/** code highlight */

.cm-s-inner .cm-variable,
.cm-s-inner .cm-operator,
.cm-s-inner .cm-property {
color: #b8bfc6;
}

.cm-s-inner .cm-keyword {
color: #C88FD0;
}

.cm-s-inner .cm-tag {
color: #7DF46A;
}

.cm-s-inner .cm-attribute {
color: #7575E4;
}

.CodeMirror div.CodeMirror-cursor {
border-left: 1px solid #b8bfc6;
z-index: 3;
}

.cm-s-inner .cm-string {
color: #D26B6B;
}

.cm-s-inner .cm-comment,
.cm-s-inner.cm-comment {
color: #DA924A;
}

.cm-s-inner .cm-header,
.cm-s-inner .cm-def,
.cm-s-inner.cm-header,
.cm-s-inner.cm-def {
color: #8d8df0;
}

.cm-s-inner .cm-quote,
.cm-s-inner.cm-quote {
color: #57ac57;
}

.cm-s-inner .cm-hr {
color: #d8d5d5;
}

.cm-s-inner .cm-link {
color: #d3d3ef;
}

.cm-s-inner .cm-negative {
color: #d95050;
}

.cm-s-inner .cm-positive {
color: #50e650;
}

.cm-s-inner .cm-string-2 {
color: #f50;
}

.cm-s-inner .cm-meta,
.cm-s-inner .cm-qualifier {
color: #b7b3b3;
}

.cm-s-inner .cm-builtin {
color: #f3b3f8;
}

.cm-s-inner .cm-bracket {
color: #997;
}

.cm-s-inner .cm-atom,
.cm-s-inner.cm-atom {
color: #84B6CB;
}

.cm-s-inner .cm-number {
color: #64AB8F;
}

.cm-s-inner .cm-variable {
color: #b8bfc6;
}

.cm-s-inner .cm-variable-2 {
color: #9FBAD5;
}

.cm-s-inner .cm-variable-3 {
color: #1cc685;
}

.CodeMirror-selectedtext,
.CodeMirror-selected {
background: #4a89dc;
color: #fff !important;
text-shadow: none;
}

.CodeMirror-gutters {
border-right: none;
}

/* CSS Document */

/** markdown source **/
.cm-s-typora-default .cm-header,
.cm-s-typora-default .cm-property
{
color: #cebcca;
}

.CodeMirror.cm-s-typora-default div.CodeMirror-cursor{
border-left: 3px solid #b8bfc6;
}

.cm-s-typora-default .cm-comment {
color: #9FB1FF;
}

.cm-s-typora-default .cm-string {
color: #A7A7D9
}

.cm-s-typora-default .cm-atom, .cm-s-typora-default .cm-number {
color: #848695;
font-style: italic;
}

.cm-s-typora-default .cm-link {
color: #95B94B;
}

.cm-s-typora-default .CodeMirror-activeline-background {
background: rgba(51, 51, 51, 0.72);
}

.cm-s-typora-default .cm-comment, .cm-s-typora-default .cm-code {
color: #8aa1e1;
}@import "";
@import "";
@import "";

:root {
--bg-color: #363B40;
--side-bar-bg-color: #2E3033;
--text-color: #b8bfc6;

--select-text-bg-color:#4a89dc;

--item-hover-bg-color: #0a0d16;
--control-text-color: #b7b7b7;
--control-text-hover-color: #eee;
--window-border: 1px solid #555;

--active-file-bg-color: rgb(34, 34, 34);
--active-file-border-color: #8d8df0;

--primary-color: #a3d5fe;

--active-file-text-color: white;
--item-hover-bg-color: #70717d;
--item-hover-text-color: white;
--primary-color: #6dc1e7;

--rawblock-edit-panel-bd: #333;

--search-select-bg-color: #428bca;
}

html {
font-size: 16px;
-webkit-font-smoothing: antialiased;
}

html,
body {
-webkit-text-size-adjust: 100%;
-ms-text-size-adjust: 100%;
background: #363B40;
background: var(--bg-color);
fill: currentColor;
line-height: 1.625rem;
}

#write {
max-width: 914px;
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@media only screen and (min-width: 1400px) {
#write {
max-width: 1024px;
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@media only screen and (min-width: 1800px) {
#write {
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html,
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textarea,
div.code-tooltip-content {
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border-color: transparent;
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div.code-tooltip,
.md-hover-tip .md-arrow:after {
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.native-window #md-notification {
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.popover.bottom > .arrow:after {
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color: #DEDEDE
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/* 36px */
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/* 40px */
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/* 24px */
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h2 {
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/* 24px */
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/* 30px */
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/* 16px */
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font-size: 0.97rem;
/* 16px */
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h3.md-focus:before,
h4.md-focus:before,
h5.md-focus:before,
h6.md-focus:before {
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position: absolute;
left: -1.642857143rem;
top: .357142857rem;
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h6.md-focus:before {
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a:focus {
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.wp-video-shortcode {
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margin-bottom: 1.5rem;
/* 24px */
}

audio:not([controls]) {
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[hidden] {
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::-moz-selection {
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*.in-text-selection,
::selection {
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color: #fff;
text-shadow: none;
}

ul,
ol {
padding: 0 0 0 1.875rem;
/* 30px */
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ol {
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ul ul,
ol ol,
ul ol,
ol ul {
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b,
th,
dt,
strong {
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}

i,
em,
dfn,
cite {
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blockquote {
padding-left: 1.875rem;
margin: 0 0 1.875rem 1.875rem;
border-left: solid 2px #474d54;
padding-left: 30px;
margin-top: 35px;
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pre,
code,
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tt,
var {
font-size: 0.875em;
font-family: Monaco, Consolas, "Andale Mono", "DejaVu Sans Mono", monospace;
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code,
tt,
var {
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kbd {
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background-color: #333;
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box-shadow: inset 0 -1px 0 rgba(0,0,0,.25);
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pre.md-fences {
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margin-bottom: 20px;
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.CodeMirror-gutters {
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border-right: 1px solid transparent;
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.enable-diagrams pre.md-fences[lang="sequence"] .code-tooltip,
.enable-diagrams pre.md-fences[lang="flow"] .code-tooltip,
.enable-diagrams pre.md-fences[lang="mermaid"] .code-tooltip {
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right: 4px;
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tt,
var {
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table {
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th,
td {
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a {
-webkit-transition: all .2s ease-in-out;
transition: all .2s ease-in-out;
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hr {
background: #474d54;
/* variable */
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h1 {
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a {
color: #e0e0e0;
text-decoration: underline;
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a:hover {
color: #fff;
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.md-inline-math script {
color: #81b1db;
}

b,
th,
dt,
strong {
color: #DEDEDE;
/* variable */
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mark {
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}

blockquote {
color: #9DA2A6;
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table a {
color: #DEDEDE;
/* variable */
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th,
td {
border: solid 1px #474d54;
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.task-list {
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.md-task-list-item > input:before {
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vertical-align: middle;
text-align: center;
border: 1px solid #b8bfc6;
background-color: #363B40;
margin-top: -0.4rem;
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.md-task-list-item > input:checked:before,
.md-task-list-item > input[checked]:before {
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/*◘*/
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.btn,
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color: #b8bfc6;
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.ty-table-edit {
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background-color: #363B40;
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</style><title>FFT - 快速傅里叶变换</title>
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<body class='typora-export os-windows'><div class='typora-export-content'>
<div id='write' class=''><h1 id='fft---快速傅里叶变换'><span>FFT - 快速傅里叶变换</span></h1><div class='md-toc' mdtype='toc'><p class="md-toc-content" role="list"><span role="listitem" class="md-toc-item md-toc-h1" data-ref="n0"><a class="md-toc-inner" href="#fft---快速傅里叶变换">FFT - 快速傅里叶变换</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n2"><a class="md-toc-inner" href="#写在前面">写在前面</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n4"><a class="md-toc-inner" href="#目的">目的</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n8"><a class="md-toc-inner" href="#前置知识">前置知识</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n9"><a class="md-toc-inner" href="#原根">原根</a></span><span role="listitem" class="md-toc-item md-toc-h3" data-ref="n11"><a class="md-toc-inner" href="#单位根">单位根</a></span><span role="listitem" class="md-toc-item md-toc-h3" data-ref="n18"><a class="md-toc-inner" href="#单位根性质">单位根性质</a></span><span role="listitem" class="md-toc-item md-toc-h3" data-ref="n21"><a class="md-toc-inner" href="#等比数列求和公式">等比数列求和公式</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n25"><a class="md-toc-inner" href="#正文">正文</a></span><span role="listitem" class="md-toc-item md-toc-h3" data-ref="n26"><a class="md-toc-inner" href="#单位根反演">单位根反演</a></span><span role="listitem" class="md-toc-item md-toc-h3" data-ref="n35"><a class="md-toc-inner" href="#推式子">推式子</a></span><span role="listitem" class="md-toc-item md-toc-h3" data-ref="n54"><a class="md-toc-inner" href="#继续推式子">继续推式子</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n72"><a class="md-toc-inner" href="#code">Code</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n137"><a class="md-toc-inner" href="#优化">优化</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n141"><a class="md-toc-inner" href="#写在后面">写在后面</a></span></p></div><p> </p><h2 id='写在前面'><span>写在前面</span></h2><p><span>该博客仅为个人对一些算法的理解与总结，不保证正确性，同时欢迎各位纠正。</span></p><h2 id='目的'><span>目的</span></h2><p><span>FFT (Fast Fourier Transform) 是为了为快速求出两个多项式的卷积，也就是 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="19.45ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 8597 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-78-TEX-I-1D436" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 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xlink:href="#MJX-80-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(6819.3,0)"><use data-c="29" xlink:href="#MJX-80-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex"> A(x), B(x), C(x) </script><span> 为多项式 )</span></p><h2 id='前置知识'><span>前置知识</span></h2><h2 id='原根'><span>原根</span></h2><p><span>详细定义可参考 </span><a href='https://zhuanlan.zhihu.com/p/166043237'><span>知乎</span></a><span> 或 </span><a href='https://oi-wiki.org/math/number-theory/primitive-root/'><span>OI-WIKI</span></a><span>，简而言之就是，对于模 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.986ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 878 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-89-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-89-TEX-I-1D45A"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> m </script><span> 意义下的原根 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.197ex" height="1.02ex" role="img" focusable="false" viewBox="0 -441 529 451" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-82-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-82-TEX-I-1D44E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> a </script><span>，有 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="23.013ex" height="2.46ex" role="img" focusable="false" viewBox="0 -893.3 10171.7 1087.3" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-83-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-83-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 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64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path id="MJX-83-TEX-N-6F" d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z"></path><path id="MJX-83-TEX-N-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z"></path></defs><g 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transform="translate(4159.1,0)"><use data-c="2C" xlink:href="#MJX-83-TEX-N-2C"></use></g><g data-mml-node="msup" transform="translate(4603.8,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-83-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D711" xlink:href="#MJX-83-TEX-I-1D711"></use></g><g data-mml-node="mo" transform="translate(654,0)"><use data-c="28" xlink:href="#MJX-83-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1043,0)"><use data-c="1D45A" xlink:href="#MJX-83-TEX-I-1D45A"></use></g><g data-mml-node="mo" transform="translate(1921,0)"><use data-c="29" xlink:href="#MJX-83-TEX-N-29"></use></g></g></g><g data-mml-node="mo" transform="translate(7127,0)"><use data-c="6D" xlink:href="#MJX-83-TEX-N-6D"></use><use data-c="6F" xlink:href="#MJX-83-TEX-N-6F" transform="translate(833,0)"></use><use data-c="64" xlink:href="#MJX-83-TEX-N-64" transform="translate(1333,0)"></use></g><g data-mml-node="mi" transform="translate(9293.7,0)"><use data-c="1D45A" xlink:href="#MJX-83-TEX-I-1D45A"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>1</mn></msup><mo>,</mo><msup><mi>a</mi><mn>2</mn></msup><mo>,</mo><mo>⋯</mo><mo>,</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>φ</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup><mo>mod</mo><mi>m</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> a^1, a^2, \cdots, a^{\varphi(m)} \bmod m </script><span> 的值各不相同，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.226ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2310 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path 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367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-84-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D711" xlink:href="#MJX-84-TEX-I-1D711"></use></g><g data-mml-node="mo" transform="translate(654,0)"><use data-c="28" xlink:href="#MJX-84-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1043,0)"><use data-c="1D45A" xlink:href="#MJX-84-TEX-I-1D45A"></use></g><g data-mml-node="mo" transform="translate(1921,0)"><use data-c="29" xlink:href="#MJX-84-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \varphi(m) </script><span> 表示 </span><a href='https://oi-wiki.org/math/number-theory/euler/'><span>欧拉函数</span></a><span> 。</span></p><h3 id='单位根'><span>单位根</span></h3><p><span>对于 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.042ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5764.4 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-85-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path><path id="MJX-85-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-85-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-85-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-85-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-85-TEX-N-2260" d="M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z"></path><path id="MJX-85-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-85-TEX-I-1D716"></use></g><g data-mml-node="mi" transform="translate(439,363) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-85-TEX-I-1D45B"></use></g></g><g data-mml-node="mo" transform="translate(1191,0)"><use data-c="3D" xlink:href="#MJX-85-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2246.8,0)"><use data-c="31" xlink:href="#MJX-85-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(2746.8,0)"><use data-c="28" xlink:href="#MJX-85-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(3135.8,0)"><use data-c="1D716" xlink:href="#MJX-85-TEX-I-1D716"></use></g><g data-mml-node="mo" transform="translate(3819.6,0)"><use data-c="2260" xlink:href="#MJX-85-TEX-N-2260"></use></g><g data-mml-node="mn" transform="translate(4875.4,0)"><use data-c="31" xlink:href="#MJX-85-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(5375.4,0)"><use data-c="29" xlink:href="#MJX-85-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ϵ</mi><mi>n</mi></msup><mo>=</mo><mn>1</mn><mo stretchy="false">(</mo><mi>ϵ</mi><mo>≠</mo><mn>1</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \epsilon^n = 1 (\epsilon \neq 1)</script><span>，称 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.919ex" height="1ex" role="img" focusable="false" viewBox="0 -431 406 442" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-86-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-86-TEX-I-1D716"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\epsilon</script><span> 为 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1393-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-1393-TEX-I-1D45B"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> n </script><span> 次单位根，其可以为模意义下的或复数意义下的。</span></p><p><span>对于模 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.986ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 878 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-89-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-89-TEX-I-1D45A"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> m </script><span> 意义下的， </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.986ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 878 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-89-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-89-TEX-I-1D45A"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> m </script><span> 为质数，令原根为 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.079ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 477 647" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-90-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D454" xlink:href="#MJX-90-TEX-I-1D454"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> g </script><span> 则有 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.374ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5911.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-91-TEX-N-67" d="M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z"></path><path id="MJX-91-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-91-TEX-N-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z"></path><path id="MJX-91-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-91-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 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73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-91-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-91-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-91-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="67" xlink:href="#MJX-91-TEX-N-67"></use><use data-c="63" xlink:href="#MJX-91-TEX-N-63" transform="translate(500,0)"></use><use data-c="64" xlink:href="#MJX-91-TEX-N-64" transform="translate(944,0)"></use></g><g data-mml-node="mo" transform="translate(1500,0)"><use data-c="28" xlink:href="#MJX-91-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1889,0)"><use data-c="1D454" xlink:href="#MJX-91-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(2366,0)"><use data-c="2C" xlink:href="#MJX-91-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2810.7,0)"><use data-c="1D45A" xlink:href="#MJX-91-TEX-I-1D45A"></use></g><g data-mml-node="mo" transform="translate(3688.7,0)"><use data-c="29" xlink:href="#MJX-91-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4355.4,0)"><use data-c="3D" xlink:href="#MJX-91-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(5411.2,0)"><use data-c="31" xlink:href="#MJX-91-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="OP" movablelimits="true">gcd</mo><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \gcd(g, m) = 1 </script><span>，此时若满足 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.127ex" height="2.26ex" role="img" focusable="false" viewBox="0 -750 4034 999" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.563ex;"><defs><path id="MJX-92-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-92-TEX-N-2223" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path id="MJX-92-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-92-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-92-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 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type="math/tex"> n \mid m - 1 </script><span> 则有 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.345ex" height="2.675ex" role="img" focusable="false" viewBox="0 -977.4 3688.7 1182.4" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-93-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path><path id="MJX-93-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-93-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 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648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-93-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-93-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-93-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 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data-mml-node="mrow" transform="translate(220,398) scale(0.707)"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-93-TEX-I-1D45A"></use></g><g data-mml-node="mo" transform="translate(878,0)"><use data-c="2212" xlink:href="#MJX-93-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1656,0)"><use data-c="31" xlink:href="#MJX-93-TEX-N-31"></use></g></g><g data-mml-node="mi" transform="translate(770.1,-345) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-93-TEX-I-1D45B"></use></g><rect width="1724.5" height="60" x="120" y="220"></rect></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi><mo>=</mo><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></mfrac></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \epsilon = g^{\frac{m - 1}{n}} </script><span>。</span></p><p><span>证明：</span></p><div 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transform="translate(1656,0)"><use data-c="31" xlink:href="#MJX-1364-TEX-N-31"></use></g></g></g></g><g data-mml-node="mtd" transform="translate(7648.4,0)"><g data-mml-node="mo"><use data-c="2261" xlink:href="#MJX-1364-TEX-N-2261"></use></g><g data-mml-node="mn" transform="translate(1055.8,0)"><use data-c="31" xlink:href="#MJX-1364-TEX-N-31"></use></g><g data-mml-node="mspace" transform="translate(1555.8,0)"></g><g data-mml-node="mo" transform="translate(2555.8,0)"><use data-c="28" xlink:href="#MJX-1364-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2944.8,0)"><use data-c="6D" xlink:href="#MJX-1364-TEX-N-6D"></use><use data-c="6F" xlink:href="#MJX-1364-TEX-N-6F" transform="translate(833,0)"></use><use data-c="64" xlink:href="#MJX-1364-TEX-N-64" transform="translate(1333,0)"></use></g><g data-mml-node="mspace" transform="translate(4833.8,0)"></g><g data-mml-node="mi" transform="translate(5333.4,0)"><use data-c="1D45A" xlink:href="#MJX-1364-TEX-I-1D45A"></use></g><g data-mml-node="mo" transform="translate(6211.4,0)"><use data-c="29" xlink:href="#MJX-1364-TEX-N-29"></use></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd></mtd><mtd><mo stretchy="false">(</mo><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></mfrac></mrow></msup><msup><mo stretchy="false">)</mo><mi>n</mi></msup></mtd><mtd><mo>≡</mo><mn>1</mn><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>m</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mo stretchy="false">⟺</mo><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle></mtd><mtd><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd><mtd><mo>≡</mo><mn>1</mn><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>m</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>由 </span><a href='https://oi-wiki.org/math/number-theory/fermat/'><span>费马小定理</span></a><span> 可知显然成立</span></p><p><span>对于复数意义下的，则可将一单位圆 n 等分，并取该 n 个点表示的复数，从 x 轴，也就是从 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.029ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2222.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-94-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-94-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-94-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-94-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 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display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex"> (1, 0) </script><span> 开始取，逆时针从 0 开始对这些复数进行编号，对于第 k 个复数记作 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.066ex" height="2.508ex" role="img" focusable="false" viewBox="0 -853.7 913.3 1108.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.576ex;"><defs><path id="MJX-212-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path><path id="MJX-212-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-212-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msubsup"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-212-TEX-I-1D716"></use></g><g data-mml-node="mi" transform="translate(439,363) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-212-TEX-I-1D458"></use></g><g data-mml-node="mi" transform="translate(439,-247) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-212-TEX-I-1D45B"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ϵ</mi><mi>n</mi><mi>k</mi></msubsup></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \epsilon_n^k </script><span>，则对于幅角非零且最小的复数 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.066ex" height="2.463ex" role="img" focusable="false" viewBox="0 -833.9 913.3 1088.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.576ex;"><defs><path id="MJX-221-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path><path id="MJX-221-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 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xlink:href="#MJX-221-TEX-I-1D716"></use></g><g data-mml-node="mn" transform="translate(439,363) scale(0.707)"><use data-c="31" xlink:href="#MJX-221-TEX-N-31"></use></g><g data-mml-node="mi" transform="translate(439,-247) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-221-TEX-I-1D45B"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ϵ</mi><mi>n</mi><mn>1</mn></msubsup></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \epsilon_n^1 </script><span>，由复数相乘时模长相乘幅角相加可知一定有 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.931ex" height="2.508ex" role="img" focusable="false" viewBox="0 -853.7 4389.5 1108.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.576ex;"><defs><path id="MJX-220-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 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transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-220-TEX-I-1D716"></use></g><g data-mml-node="mn" transform="translate(439,363) scale(0.707)"><use data-c="31" xlink:href="#MJX-220-TEX-N-31"></use></g><g data-mml-node="mi" transform="translate(439,-247) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-220-TEX-I-1D45B"></use></g></g><g data-mml-node="msup" transform="translate(1302.3,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-220-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(422,363) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-220-TEX-I-1D458"></use></g></g><g data-mml-node="mo" transform="translate(2420.4,0)"><use data-c="3D" xlink:href="#MJX-220-TEX-N-3D"></use></g><g data-mml-node="msubsup" transform="translate(3476.2,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-220-TEX-I-1D716"></use></g><g data-mml-node="mi" transform="translate(439,363) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-220-TEX-I-1D458"></use></g><g data-mml-node="mi" transform="translate(439,-247) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-220-TEX-I-1D45B"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msubsup><mi>ϵ</mi><mi>n</mi><mn>1</mn></msubsup><msup><mo stretchy="false">)</mo><mi>k</mi></msup><mo>=</mo><msubsup><mi>ϵ</mi><mi>n</mi><mi>k</mi></msubsup></math></mjx-assistive-mml></mjx-container><script type="math/tex"> (\epsilon_n^1)^k = \epsilon_n^k </script><span>，则称 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.066ex" height="2.463ex" role="img" focusable="false" viewBox="0 -833.9 913.3 1088.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.576ex;"><defs><path id="MJX-221-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 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151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msubsup"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-221-TEX-I-1D716"></use></g><g data-mml-node="mn" transform="translate(439,363) scale(0.707)"><use data-c="31" xlink:href="#MJX-221-TEX-N-31"></use></g><g data-mml-node="mi" transform="translate(439,-247) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-221-TEX-I-1D45B"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ϵ</mi><mi>n</mi><mn>1</mn></msubsup></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \epsilon_n^1 </script><span> 为 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1393-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-1393-TEX-I-1D45B"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> n </script><span> 次单位根。</span></p><p><span>很多地方可能用 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.407ex" height="1.027ex" role="img" focusable="false" viewBox="0 -443 622 454" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-227-TEX-I-1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-227-TEX-I-1D714"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \omega </script><span> 来表示单位根，也就是本文中的 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.919ex" height="1ex" role="img" focusable="false" viewBox="0 -431 406 442" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-245-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-245-TEX-I-1D716"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \epsilon </script><span>，仅为表示方式的区别而已。</span></p><h3 id='单位根性质'><span>单位根性质</span></h3><p><span>对于 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1393-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-1393-TEX-I-1D45B"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> n </script><span> 次单位根有如下式子</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n124" cid="n124" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="15.99ex" height="4.897ex" role="img" focusable="false" viewBox="0 -1914.5 7067.6 2164.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;" class=""><defs><path id="MJX-1457-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path 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220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-1457-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-1457-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-1457-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1457-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-1457-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-1457-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-1457-TEX-I-1D716"></use></g><g data-mml-node="mi" transform="translate(439,413) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-1457-TEX-I-1D458"></use></g></g><g data-mml-node="msup" transform="translate(1246.4,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-1457-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-1457-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(2349.7,0)"><use data-c="3D" xlink:href="#MJX-1457-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(3405.5,0)"><use data-c="28" xlink:href="#MJX-1457-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(3794.5,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-1457-TEX-I-1D716"></use></g><g data-mml-node="TeXAtom" transform="translate(439,796.5) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-1457-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2B" xlink:href="#MJX-1457-TEX-N-2B"></use></g><g data-mml-node="mstyle" transform="translate(1299,0) scale(1.414)"><g data-mml-node="mfrac"><g data-mml-node="mi" transform="translate(220,676)"><use data-c="1D45B" xlink:href="#MJX-1457-TEX-I-1D45B"></use></g><g data-mml-node="mn" transform="translate(270,-686)"><use data-c="32" xlink:href="#MJX-1457-TEX-N-32"></use></g><rect width="800" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="msup" transform="translate(6242,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-1457-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-1457-TEX-N-32"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mi>k</mi></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>n</mi><mn>2</mn></mfrac></mstyle></mrow></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></mjx-assistive-mml></mjx-container></div></div><p><span>证明</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n128" cid="n128" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-rawblock-container md-math-container" contenteditable="false" 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xlink:href="#MJX-1532-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(1333.6,0)"><use data-c="28" xlink:href="#MJX-1532-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(1722.6,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-1532-TEX-I-1D716"></use></g><g data-mml-node="mi" transform="translate(439,413) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-1532-TEX-I-1D458"></use></g></g><g data-mml-node="msup" transform="translate(2580,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-1532-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-1532-TEX-N-32"></use></g></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>n</mi><mn>2</mn></mfrac></mstyle></mrow></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mi>k</mi></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>×</mo><msup><mi>ϵ</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mi>k</mi></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p> </p><h3 id='等比数列求和公式'><span>等比数列求和公式</span></h3><p><a href='https://baike.baidu.com/item/%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%E6%B1%82%E5%92%8C%E5%85%AC%E5%BC%8F/7527367'><span>详细证明</span></a></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n23" cid="n23" mdtype="math_block" data-math-tag-before="0" 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transform="translate(1398,0)"><use data-c="3D" xlink:href="#MJX-1367-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(2453.8,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-1367-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(562,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1367-TEX-N-31"></use></g></g><g data-mml-node="mstyle" transform="translate(3419.4,0)"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-1367-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2212" xlink:href="#MJX-1367-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(1722.4,0)"><g data-mml-node="mi"><use data-c="1D45E" xlink:href="#MJX-1367-TEX-I-1D45E"></use></g><g data-mml-node="mi" transform="translate(543.7,363) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-1367-TEX-I-1D45B"></use></g></g></g><g data-mml-node="mrow" transform="translate(499,-686)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-1367-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2212" xlink:href="#MJX-1367-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(1722.4,0)"><use data-c="1D45E" xlink:href="#MJX-1367-TEX-I-1D45E"></use></g></g><rect width="2940.4" height="60" x="120" y="220"></rect></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mi>q</mi><mi>n</mi></msup></mrow><mrow><mn>1</mn><mo>−</mo><mi>q</mi></mrow></mfrac></mstyle></math></mjx-assistive-mml></mjx-container></div></div><h2 id='正文'><span>正文</span></h2><h3 id='单位根反演'><span>单位根反演</span></h3><p><span>对于 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1393-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-1393-TEX-I-1D45B"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> n </script><span> 次单位根 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.919ex" height="1ex" role="img" focusable="false" viewBox="0 -431 406 442" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-245-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-245-TEX-I-1D716"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \epsilon </script><span>，显然有如下的， </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="11.256ex" height="2.226ex" role="img" focusable="false" viewBox="0 -833.9 4975.2 983.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-102-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 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xlink:href="#MJX-102-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(2299.1,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-102-TEX-I-1D716"></use></g><g data-mml-node="mn" transform="translate(439,363) scale(0.707)"><use data-c="30" xlink:href="#MJX-102-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(3419.4,0)"><use data-c="3D" xlink:href="#MJX-102-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(4475.2,0)"><use data-c="31" xlink:href="#MJX-102-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><msup><mi>ϵ</mi><mn>0</mn></msup><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex"> a_1 = \epsilon^0 = 1 </script><span>， </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.94ex" 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xlink:href="#MJX-1368-TEX-I-1D716"></use></g><g data-mml-node="mi" transform="translate(439,289) scale(0.707)"><use data-c="1D463" xlink:href="#MJX-1368-TEX-I-1D463"></use></g></g></g><rect width="3178.7" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mo" transform="translate(4752.2,0)"><use data-c="28" xlink:href="#MJX-1368-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(5141.2,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-1368-TEX-I-1D716"></use></g><g data-mml-node="mi" transform="translate(439,413) scale(0.707)"><use data-c="1D463" xlink:href="#MJX-1368-TEX-I-1D463"></use></g></g><g data-mml-node="mo" transform="translate(6250.9,0)"><use data-c="2260" xlink:href="#MJX-1368-TEX-N-2260"></use></g><g data-mml-node="mn" transform="translate(7306.7,0)"><use data-c="31" xlink:href="#MJX-1368-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(7806.7,0)"><use data-c="29" xlink:href="#MJX-1368-TEX-N-29"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-1633.4)"><g data-mml-node="mtd" transform="translate(2686.6,0)"></g><g data-mml-node="mtd" transform="translate(2686.6,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-1368-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="30" xlink:href="#MJX-1368-TEX-N-30"></use></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi></mrow></msup></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>n</mi><mi>v</mi></mrow></msup></mrow><mrow><mn>1</mn><mo>−</mo><msup><mi>ϵ</mi><mi>v</mi></msup></mrow></mfrac></mstyle><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mi>v</mi></msup><mo>≠</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>又有 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-1393-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-1393-TEX-I-1D45B"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> n </script><span> 次单位根性质可知 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.031ex" height="1.715ex" role="img" focusable="false" viewBox="0 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data-mml-node="mn" transform="translate(2165.5,0)"><use data-c="31" xlink:href="#MJX-105-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ϵ</mi><mi>v</mi></msup><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \epsilon^v = 1 </script><span> 时，上式 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.52ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 1555.8 748" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-106-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-106-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 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157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-1369-TEX-N-2223" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-1369-TEX-I-1D716"></use></g><g data-mml-node="mi" transform="translate(439,413) scale(0.707)"><use data-c="1D463" xlink:href="#MJX-1369-TEX-I-1D463"></use></g></g><g data-mml-node="mo" transform="translate(1109.7,0)"><use data-c="3D" xlink:href="#MJX-1369-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2165.5,0)"><use data-c="31" xlink:href="#MJX-1369-TEX-N-31"></use></g><g data-mml-node="mstyle" transform="translate(2665.5,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(3221.3,0)"><use data-c="27FA" xlink:href="#MJX-1369-TEX-N-27FA"></use></g><g data-mml-node="mstyle" transform="translate(5079.3,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(5635.1,0)"><use data-c="1D45B" xlink:href="#MJX-1369-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(6512.8,0)"><use data-c="2223" xlink:href="#MJX-1369-TEX-N-2223"></use></g><g data-mml-node="mi" transform="translate(7068.6,0)"><use data-c="1D463" xlink:href="#MJX-1369-TEX-I-1D463"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>ϵ</mi><mi>v</mi></msup><mo>=</mo><mn>1</mn><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mo stretchy="false">⟺</mo><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mi>n</mi><mo>∣</mo><mi>v</mi></math></mjx-assistive-mml></mjx-container></div></div><p><span>综上则有如下式子</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n33" cid="n33" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="23.562ex" height="6.712ex" role="img" focusable="false" viewBox="0 -1733.4 10414.3 2966.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.791ex;" class=""><defs><path id="MJX-1370-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 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data-mml-node="mtr" transform="translate(0,701)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-1370-TEX-N-30"></use></g></g><g data-mml-node="mtd" transform="translate(1500,0)"><g data-mml-node="mstyle"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(1000,0)"><use data-c="1D45B" xlink:href="#MJX-1370-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(1877.8,0)"><use data-c="2224" xlink:href="#MJX-1370-TEX-N-2224"></use></g><g data-mml-node="mi" transform="translate(2452.6,0)"><use data-c="1D463" xlink:href="#MJX-1370-TEX-I-1D463"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-701)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-1370-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(1500,0)"><g data-mml-node="mstyle"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(1000,0)"><use data-c="1D45B" xlink:href="#MJX-1370-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(1877.8,0)"><use data-c="2223" xlink:href="#MJX-1370-TEX-N-2223"></use></g><g data-mml-node="mi" transform="translate(2433.6,0)"><use data-c="1D463" xlink:href="#MJX-1370-TEX-I-1D463"></use></g></g></g></g><g data-mml-node="mo" transform="translate(5187.6,0) translate(0 250)"></g></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi></mrow></msup><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mi>n</mi><mo>∤</mo><mi>v</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mi>n</mi><mo>∣</mo><mi>v</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>此即为单位根反演</span></p><h3 id='推式子'><span>推式子</span></h3><p><span>将单位根反演代入原式，令 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="12.605ex" height="1.934ex" role="img" focusable="false" viewBox="0 -661 5571.4 855" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-107-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 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stretchy="false">)</mo><mo>×</mo><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>p</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>q</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>a</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>×</mo><mi>b</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow 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data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>p</mi></mrow></msup><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>q</mi></mrow></msup><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>k</mi><mi>i</mi></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>p</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>a</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>p</mi></mrow></msup><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>q</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>b</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>q</mi></mrow></msup><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>k</mi><mi>i</mi></mrow></msup></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle><mo stretchy="false">⟺</mo><mstyle scriptlevel="0"><mspace width="0.278em"></mspace></mstyle></mtd><mtd><mi>n</mi><mo>×</mo><mi>c</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>i</mi></mrow></msup><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>p</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>a</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>p</mi></mrow></msup><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>q</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>b</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>q</mi></mrow></msup></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>观察最后两个式子，可以发现如下两个式子</span></p><div contenteditable="false" 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display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>i</mi></mrow></msup><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>g</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>p</mi><mi>i</mi></mrow></msup><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>证明</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n44" cid="n44" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.181ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4058 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;" class=""><defs><path id="MJX-1375-TEX-N-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path><path id="MJX-1375-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-1375-TEX-I-1D442" d="M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435ZM637 476Q637 565 591 615T476 665Q396 665 322 605Q242 542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z"></path><path id="MJX-1375-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2F" xlink:href="#MJX-1375-TEX-N-2F"></use></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(500,0)"><g data-mml-node="mo"><use data-c="2F" xlink:href="#MJX-1375-TEX-N-2F"></use></g></g><g data-mml-node="mi" transform="translate(1000,0)"><use data-c="1D447" xlink:href="#MJX-1375-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(1704,0)"><use data-c="1D442" xlink:href="#MJX-1375-TEX-I-1D442"></use></g><g data-mml-node="mi" transform="translate(2467,0)"><use data-c="1D437" xlink:href="#MJX-1375-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(3295,0)"><use data-c="1D442" xlink:href="#MJX-1375-TEX-I-1D442"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>T</mi><mi>O</mi><mi>D</mi><mi>O</mi></math></mjx-assistive-mml></mjx-container></div></div><p><span>则可知求 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.143ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1831 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-116-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-116-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-116-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-116-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D453" xlink:href="#MJX-116-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(550,0)"><use data-c="28" xlink:href="#MJX-116-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(939,0)"><use data-c="1D45D" xlink:href="#MJX-116-TEX-I-1D45D"></use></g><g data-mml-node="mo" transform="translate(1442,0)"><use data-c="29" xlink:href="#MJX-116-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex"> f(p) </script><span> 的过程即为DFT，求 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.977ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1758 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-113-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path id="MJX-113-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-113-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-113-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D454" xlink:href="#MJX-113-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(477,0)"><use data-c="28" xlink:href="#MJX-113-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(866,0)"><use data-c="1D45D" xlink:href="#MJX-113-TEX-I-1D45D"></use></g><g data-mml-node="mo" transform="translate(1369,0)"><use data-c="29" xlink:href="#MJX-113-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex"> g(p) </script><span> 的过程即为IDFT。</span></p><p><span>由定义显然有</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n47" cid="n47" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="37.038ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 16371 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;" class=""><defs><path id="MJX-1376-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path><path id="MJX-1376-TEX-I-1D439" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></path><path id="MJX-1376-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path id="MJX-1376-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-1376-TEX-I-1D436" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path><path id="MJX-1376-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-1376-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-1376-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-1376-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1376-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-1376-TEX-N-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path><path id="MJX-1376-TEX-I-1D435" d="M231 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transform="translate(1577,0)"><use data-c="1D447" xlink:href="#MJX-1376-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(2281,0)"><use data-c="28" xlink:href="#MJX-1376-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2670,0)"><use data-c="1D436" xlink:href="#MJX-1376-TEX-I-1D436"></use></g><g data-mml-node="mo" transform="translate(3430,0)"><use data-c="2C" xlink:href="#MJX-1376-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(3874.7,0)"><use data-c="1D456" xlink:href="#MJX-1376-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(4219.7,0)"><use data-c="29" xlink:href="#MJX-1376-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4886.4,0)"><use data-c="3D" xlink:href="#MJX-1376-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(5942.2,0)"><use data-c="1D437" xlink:href="#MJX-1376-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(6770.2,0)"><use data-c="1D439" xlink:href="#MJX-1376-TEX-I-1D439"></use></g><g data-mml-node="mi" transform="translate(7519.2,0)"><use data-c="1D447" xlink:href="#MJX-1376-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(8223.2,0)"><use data-c="28" xlink:href="#MJX-1376-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(8612.2,0)"><use data-c="1D434" xlink:href="#MJX-1376-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(9362.2,0)"><use data-c="2C" xlink:href="#MJX-1376-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(9806.9,0)"><use data-c="1D456" xlink:href="#MJX-1376-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(10151.9,0)"><use data-c="29" xlink:href="#MJX-1376-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(10763.1,0)"><use data-c="D7" xlink:href="#MJX-1376-TEX-N-D7"></use></g><g data-mml-node="mi" transform="translate(11763.3,0)"><use data-c="1D437" xlink:href="#MJX-1376-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(12591.3,0)"><use data-c="1D439" xlink:href="#MJX-1376-TEX-I-1D439"></use></g><g data-mml-node="mi" transform="translate(13340.3,0)"><use data-c="1D447" xlink:href="#MJX-1376-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(14044.3,0)"><use data-c="28" xlink:href="#MJX-1376-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(14433.3,0)"><use data-c="1D435" xlink:href="#MJX-1376-TEX-I-1D435"></use></g><g data-mml-node="mo" transform="translate(15192.3,0)"><use data-c="2C" xlink:href="#MJX-1376-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(15637,0)"><use data-c="1D456" xlink:href="#MJX-1376-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(15982,0)"><use data-c="29" xlink:href="#MJX-1376-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>D</mi><mi>F</mi><mi>T</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mi>F</mi><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo><mo>×</mo><mi>D</mi><mi>F</mi><mi>T</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>（ </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.146ex" height="2.059ex" role="img" focusable="false" viewBox="0 -716 3158.3 910" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-114-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-114-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-114-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 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transform="translate(750,0)"><use data-c="2C" xlink:href="#MJX-114-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(1194.7,0)"><use data-c="1D435" xlink:href="#MJX-114-TEX-I-1D435"></use></g><g data-mml-node="mo" transform="translate(1953.7,0)"><use data-c="2C" xlink:href="#MJX-114-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2398.3,0)"><use data-c="1D436" xlink:href="#MJX-114-TEX-I-1D436"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> A, B, C </script><span> 均代表该多项式）</span></p><p><span>又有</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n50" cid="n50" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container 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250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-1377-TEX-I-1D436" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path><path id="MJX-1377-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-1377-TEX-N-3D" d="M56 347Q56 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data-c="1D439" xlink:href="#MJX-1377-TEX-I-1D439"></use></g><g data-mml-node="mi" transform="translate(4751,0)"><use data-c="1D447" xlink:href="#MJX-1377-TEX-I-1D447"></use></g><g data-mml-node="mo" transform="translate(5455,0)"><use data-c="28" xlink:href="#MJX-1377-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(5844,0)"><use data-c="1D436" xlink:href="#MJX-1377-TEX-I-1D436"></use></g><g data-mml-node="mo" transform="translate(6604,0)"><use data-c="29" xlink:href="#MJX-1377-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(6993,0)"><use data-c="29" xlink:href="#MJX-1377-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(7659.8,0)"><use data-c="3D" xlink:href="#MJX-1377-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(8715.6,0)"><use data-c="1D436" xlink:href="#MJX-1377-TEX-I-1D436"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mi>D</mi><mi>F</mi><mi>T</mi><mo stretchy="false">(</mo><mi>D</mi><mi>F</mi><mi>T</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi></math></mjx-assistive-mml></mjx-container></div></div><p><span>证明</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n52" cid="n52" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.181ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4058 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;" class=""><defs><path id="MJX-1378-TEX-N-2F" d="M423 750Q432 750 438 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542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z"></path><path id="MJX-1378-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2F" xlink:href="#MJX-1378-TEX-N-2F"></use></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(500,0)"><g data-mml-node="mo"><use data-c="2F" xlink:href="#MJX-1378-TEX-N-2F"></use></g></g><g data-mml-node="mi" transform="translate(1000,0)"><use data-c="1D447" xlink:href="#MJX-1378-TEX-I-1D447"></use></g><g data-mml-node="mi" transform="translate(1704,0)"><use data-c="1D442" xlink:href="#MJX-1378-TEX-I-1D442"></use></g><g data-mml-node="mi" transform="translate(2467,0)"><use data-c="1D437" xlink:href="#MJX-1378-TEX-I-1D437"></use></g><g data-mml-node="mi" transform="translate(3295,0)"><use data-c="1D442" xlink:href="#MJX-1378-TEX-I-1D442"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>T</mi><mi>O</mi><mi>D</mi><mi>O</mi></math></mjx-assistive-mml></mjx-container></div></div><p><span>则此时多项式 C 可求，但时间复杂度仍然是 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.832ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 2577.6 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-115-TEX-I-1D442" d="M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435ZM637 476Q637 565 591 615T476 665Q396 665 322 605Q242 542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z"></path><path id="MJX-115-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-115-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-115-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-115-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex"> O(n^2) </script><span> </span></p><h3 id='继续推式子'><span>继续推式子</span></h3><p><span>对于</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n56" cid="n56" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="16.887ex" height="6.712ex" role="img" focusable="false" viewBox="0 -1720.9 7463.8 2966.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.819ex;" class=""><defs><path id="MJX-1379-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-1379-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-1379-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 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61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-1379-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-1379-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-1379-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-1379-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-1379-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 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data-mml-node="mi" transform="translate(939,0)"><use data-c="1D45D" xlink:href="#MJX-1379-TEX-I-1D45D"></use></g><g data-mml-node="mo" transform="translate(1442,0)"><use data-c="29" xlink:href="#MJX-1379-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2108.8,0)"><use data-c="3D" xlink:href="#MJX-1379-TEX-N-3D"></use></g><g data-mml-node="munderover" transform="translate(3164.6,0)"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-1379-TEX-LO-2211"></use></g><g data-mml-node="TeXAtom" transform="translate(148.2,-1087.9) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-1379-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(345,0)"><use data-c="3D" xlink:href="#MJX-1379-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="30" xlink:href="#MJX-1379-TEX-N-30"></use></g></g><g data-mml-node="TeXAtom" transform="translate(58,1150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-1379-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(600,0)"><use data-c="2212" xlink:href="#MJX-1379-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1378,0)"><use data-c="31" xlink:href="#MJX-1379-TEX-N-31"></use></g></g></g><g data-mml-node="msup" transform="translate(4775.2,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-1379-TEX-I-1D716"></use></g><g data-mml-node="TeXAtom" transform="translate(439,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-1379-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(503,0)"><use data-c="1D456" xlink:href="#MJX-1379-TEX-I-1D456"></use></g></g></g><g data-mml-node="mi" transform="translate(5863.8,0)"><use data-c="1D454" xlink:href="#MJX-1379-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(6340.8,0)"><use data-c="28" xlink:href="#MJX-1379-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(6729.8,0)"><use data-c="1D456" xlink:href="#MJX-1379-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(7074.8,0)"><use data-c="29" xlink:href="#MJX-1379-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>i</mi></mrow></msup><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>可以考虑 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.143ex" 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stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex"> f(p) </script><span> 与 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.061ex" height="2.497ex" role="img" focusable="false" viewBox="0 -853.7 4004.8 1103.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-117-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 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310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-117-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-117-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-117-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 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display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><msup><mn>2</mn><mi>k</mi></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex"> f(p + 2^k) </script><span> 的关系，则此时我们可以假设 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.572ex" height="2.117ex" role="img" focusable="false" viewBox="0 -853.7 3788.6 935.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-118-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-118-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-118-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-118-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 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556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-118-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(877.8,0)"><use data-c="3D" xlink:href="#MJX-118-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(1933.6,0)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-118-TEX-N-32"></use></g><g data-mml-node="TeXAtom" transform="translate(533,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-118-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2B" xlink:href="#MJX-118-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-118-TEX-N-31"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex"> n = 2^{k + 1} </script><span> </span></p><p><span>且令</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n59" cid="n59" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="28.311ex" height="11.991ex" role="img" focusable="false" viewBox="0 -2900 12513.4 5300" 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data-c="2212" xlink:href="#MJX-1563-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1378,0)"><use data-c="31" xlink:href="#MJX-1563-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(2777.6,0)"><use data-c="28" xlink:href="#MJX-1563-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(3166.6,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-1563-TEX-I-1D716"></use></g><g data-mml-node="mi" transform="translate(439,413) scale(0.707)"><use data-c="1D45D" xlink:href="#MJX-1563-TEX-I-1D45D"></use></g></g><g data-mml-node="msup" transform="translate(4011.2,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-1563-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(422,413) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-1563-TEX-I-1D456"></use></g></g><g data-mml-node="msup" transform="translate(4727.2,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-1563-TEX-I-1D716"></use></g><g 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columnspacing="0em" rowspacing="3pt"><mtr><mtd><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><mi>p</mi><mo>+</mo><msup><mn>2</mn><mi>k</mi></msup></mrow></msup><msup><mo stretchy="false">)</mo><mi>i</mi></msup><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mi>p</mi></msup><msup><mo stretchy="false">)</mo><mi>i</mi></msup><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mi>k</mi></msup></mrow></msup><mi>g</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>此时显然有</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n94" cid="n94" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="12.467ex" height="2.866ex" role="img" focusable="false" viewBox="0 -883.3 5510.6 1266.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.867ex;" class=""><defs><path id="MJX-1567-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 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xlink:href="#MJX-1567-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(1333.6,0)"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-1567-TEX-I-1D716"></use></g><g data-mml-node="TeXAtom" transform="translate(439,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-1567-TEX-N-32"></use></g><g data-mml-node="TeXAtom" transform="translate(533,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-1567-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2B" xlink:href="#MJX-1567-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-1567-TEX-N-31"></use></g></g></g></g></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mi>k</mi></msup></mrow></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd><mtd><mi></mi><mo>=</mo><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msup></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>且我们已知 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.919ex" height="1ex" role="img" focusable="false" viewBox="0 -431 406 442" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-245-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-245-TEX-I-1D716"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \epsilon </script><span> 为 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.197ex" height="1.932ex" role="img" focusable="false" viewBox="0 -853.7 1855.1 853.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-700-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-700-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-700-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-700-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-700-TEX-N-32"></use></g><g data-mml-node="TeXAtom" transform="translate(533,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-700-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2B" xlink:href="#MJX-700-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-700-TEX-N-31"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex"> 2^{k + 1} </script><span> 次单位根，所以显然有</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n101" cid="n101" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.222ex" height="2.486ex" role="img" focusable="false" viewBox="0 -1016.7 3634.3 1098.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;" class=""><defs><path id="MJX-1547-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path><path id="MJX-1547-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-1547-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-1547-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-1547-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-1547-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-1547-TEX-I-1D716"></use></g><g data-mml-node="TeXAtom" transform="translate(439,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-1547-TEX-N-32"></use></g><g data-mml-node="TeXAtom" transform="translate(533,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D458" xlink:href="#MJX-1547-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(521,0)"><use data-c="2B" xlink:href="#MJX-1547-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1299,0)"><use data-c="31" xlink:href="#MJX-1547-TEX-N-31"></use></g></g></g></g></g><g data-mml-node="mo" transform="translate(2078.5,0)"><use data-c="3D" xlink:href="#MJX-1547-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3134.3,0)"><use data-c="31" xlink:href="#MJX-1547-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mrow data-mjx-texclass="ORD"><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msup><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></div></div><p><span>所以 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.628ex" height="2.212ex" role="img" focusable="false" viewBox="0 -966.7 1161.7 977.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-782-TEX-I-1D716" d="M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z"></path><path id="MJX-782-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-782-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D716" xlink:href="#MJX-782-TEX-I-1D716"></use></g><g data-mml-node="TeXAtom" transform="translate(439,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-782-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(533,363) scale(0.707)"><use data-c="1D458" xlink:href="#MJX-782-TEX-I-1D458"></use></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ϵ</mi><mrow data-mjx-texclass="ORD"><msup><mn>2</mn><mi>k</mi></msup></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex"> \epsilon^{2^k} </script><span> 为二次单位根，这里为了方便，我们用 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-788-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D462" xlink:href="#MJX-788-TEX-I-1D462"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> u </script><span> 代替。</span></p><p><span>此时可以考虑令</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n109" cid="n109" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="32.1ex" height="14.556ex" role="img" focusable="false" viewBox="0 -3466.8 14188.2 6433.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -6.712ex;" class=""><defs><path id="MJX-1543-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-1543-TEX-V-2032" d="M79 43Q73 43 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26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-1543-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-1543-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1543-TEX-LO-2211" d="M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 809 1174 838T1033 881T882 898T699 902H574H543H251L259 891Q722 258 724 252Q725 250 724 246Q721 243 460 -56L196 -356Q196 -357 407 -357Q459 -357 548 -357T676 -358Q812 -358 896 -353T1063 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data-mjx-alternate="1">′</mo></msup></mrow></msup><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mtd><mtd><mo>=</mo></mtd><mtd><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><msup><mi>n</mi><mrow data-mjx-texclass="ORD"><msup><mi></mi><mo data-mjx-alternate="1">′</mo></msup></mrow></msup></mrow></munderover><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mi>p</mi></msup><msup><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msup><mi>f</mi><mrow data-mjx-texclass="ORD"><msup><mi></mi><mo data-mjx-alternate="1">′</mo></msup></mrow></msup><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><msup><mn>2</mn><mi>k</mi></msup><mo stretchy="false">)</mo></mtd><mtd><mo>=</mo></mtd><mtd><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mi>n</mi><mrow data-mjx-texclass="ORD"><msup><mi></mi><mo data-mjx-alternate="1">′</mo></msup></mrow></msup></mrow></mrow></munderover><mo stretchy="false">(</mo><msup><mi>ϵ</mi><mi>p</mi></msup><msup><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>此时式子形式便可按相同方法继续递归，直到 </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.506ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 2433.6 748" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-1250-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-1250-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-1250-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-1250-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(877.8,0)"><use data-c="3D" xlink:href="#MJX-1250-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1933.6,0)"><use data-c="32" xlink:href="#MJX-1250-TEX-N-32"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex"> n = 2 </script><span> 时进行回溯。</span></p><h2 id='code'><span>Code</span></h2><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="cpp"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="cpp"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 10.8px; left: 4px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">//TODO</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 26px;"></div><div class="CodeMirror-gutters" style="display: none; height: 26px;"></div></div></div></pre><h2 id='优化'><span>优化</span></h2><p><span>//TODO</span></p><h2 id='写在后面'><span>写在后面</span></h2><p><span>写完之后发现依然没有很清晰的弄明白，然后发现有几个Blog写的更清晰易懂</span></p><p><a href='https://zhuanlan.zhihu.com/p/31584464'><span>一小时学会快速傅里叶变换（Fast Fourier Transform）</span></a></p><p><a href='https://www.cnblogs.com/RabbitHu/p/FFT.html'><span>小学生都能看懂的FFT！！！</span></a></p><p><span>至于几个TODO等以后再慢慢填坑吧</span></p><p> </p><p> </p><p> </p><p> </p><p> </p><p> </p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n66" cid="n66" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.036ex" height="0.036ex" role="img" focusable="false" viewBox="0 0 15.9 15.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;" class=""><defs></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"></math></mjx-assistive-mml></mjx-container></div></div><p> </p></div></div>
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标签：md,cm,变换,border,FFT,color,background,0px,傅里叶
来源： https://www.cnblogs.com/tsawke/p/16572594.html

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FFT - 快速傅里叶变换