{"id":1838,"date":"2026-06-23T10:10:36","date_gmt":"2026-06-23T02:10:36","guid":{"rendered":"https:\/\/okb385.org.cn\/?p=1838"},"modified":"2026-06-23T10:11:09","modified_gmt":"2026-06-23T02:11:09","slug":"%e5%85%8b%e7%bd%97%e5%86%85%e5%85%8b%e7%ac%a6%e5%8f%b7","status":"publish","type":"post","link":"https:\/\/okb385.org.cn\/index.php\/2026\/06\/23\/%e5%85%8b%e7%bd%97%e5%86%85%e5%85%8b%e7%ac%a6%e5%8f%b7\/","title":{"rendered":"\u514b\u7f57\u5185\u514b\u7b26\u53f7"},"content":{"rendered":"\n<!--more-->\n\n\n\n<h2 class=\"wp-block-heading\">\u52d2\u8ba9\u5fb7\u7b26\u53f7<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u5b9a\u4e49<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u5bf9$p$\u5947\u8d28\u6570\uff0c$n\\in\\Z,p\\nmid n$\u5b9a\u4e49\u52d2\u8ba9\u5fb7\u7b26\u53f7\n\\begin{align*}\n\\left(\\frac{n}{p}\\right)=\\left\\{\\begin{array}{c}\n\t1\\quad &amp;n\\equiv \\square\\mod p\\\\\n\t-1\\quad &amp;n\\not\\equiv \\square\\mod p\n\\end{array}\\right.\n\\end{align*}\n<br>\u5176\u4e2d$\\square$\u662f\u4e00\u4e2a\u5b8c\u5168\u5e73\u65b9\u6570\u3002\u82e5$(\\frac{n}{p})=1$\uff0c$n$\u88ab\u79f0\u4e3a$p$\u7684\u4e8c\u6b21\u5269\u4f59\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u539f\u59cb\u5b9a\u4e49\uff1a\\begin{align*}\\left(\\frac{n}{p}\\right)\\equiv n^{\\frac{p-1}{2}}\\mod p\\end{align*}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6027\u8d28<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$\\left(\\frac{ab}{p}\\right)=\\left(\\frac{a}{p}\\right)\\left(\\frac{b}{p}\\right)$<\/li>\n\n\n\n<li>\u82e5$a\\equiv b\\mod p$\uff0c\u5219$\\left(\\frac{a}{p}\\right)=\\left(\\frac{b}{p}\\right)$<\/li>\n\n\n\n<li>\\[\n  \\left(\\frac{-1}{p}\\right)=(-1)^{\\frac{p-1}{2}}\n  \\]<\/li>\n\n\n\n<li>\\[\\left(\\frac{2}{p}\\right)=(-1)^{\\frac{p^2-1}{8}}\\]<\/li>\n\n\n\n<li>\u4e8c\u6b21\u4e92\u53cd\u5f8b\uff1a\u8bbe$p,q$\u4e3a\u5947\u8d28\u6570\uff0c\u6709\\[\\left(\\frac{p}{q}\\right)\\cdot\\left(\\frac{q}{p}\\right)=(-1)^{\\frac{(p-1)(q-1)}{4}}\\]<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">\u96c5\u53ef\u6bd4\u7b26\u53f7<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u5b9a\u4e49<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u63a8\u5e7f\u5230\u6b63\u5947\u6570\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8bbe$m$\u662f\u4e00\u4e2a\u6b63\u5947\u6570\uff0c\u5176\u8d28\u56e0\u6570\u5206\u89e3\u5f0f\u4e3a$m=\\prod_{i=1}^s p_i$\uff0c\u5e76\u4e14$(n,m)=1$\uff0c\u90a3\u4e48\u5b9a\u4e49\\[\\left(\\frac{n}{m}\\right)=\\prod_{i=1}^s \\left(\\frac{n}{p_i}\\right)\\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u514b\u7f57\u5185\u514b\u7b26\u53f7<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u5b9a\u4e49<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u63a8\u5e7f\u5230\u6574\u6570\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8bbe$m$\u662f\u975e\u96f6\u6574\u6570\uff0c\u5177\u6709\u8d28\u56e0\u6570\u5206\u89e3\\[m=u\\cdot p_1^{e_1}\\cdots p_k^{e_k}\\]\uff0c\u5176\u4e2d$u$\u662f\u5355\u4f4d\u3002\u8bbe$n$\u662f\u4e00\u4e2a\u6574\u6570\uff0c\u5b9a\u4e49\\[\\left(\\frac{n}{m}\\right)=\\left(\\frac{n}{u}\\right)\\prod_{i=1}^k \\left(\\frac{n}{p_i}\\right)^{e_i}\\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5bf9$p=2$\u5b9a\u4e49\\begin{align*}\\left(\\frac{n}{2}\\right)=\\left\\{\\begin{array}{l}\n0&amp;\\qquad n \\text{\u662f\u5076\u6570}\\\\\n1&amp;\\qquad n\\equiv \\pm 1\\pmod8\\\\\n-1&amp;\\qquad n\\equiv \\pm 3\\pmod8\\\\\n\\end{array}\\right.\\end{align*}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\n$\\left(\\frac{n}{u}\\right)$\u5728$u=1$\u65f6\u7b49\u4e8e$1$\uff0c\u5728$u=-1$\u65f6\u5b9a\u4e49\n\\begin{align*}\n\\left(\\frac{n}{-1}\\right)=\\left\\{\\begin{array}{cc}\n-1&amp;\\qquad n&lt;0\\\\\n1&amp;\\qquad n\\ge0\n\\end{array}\\right.\n\\end{align*}\n<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7279\u522b\u5730\uff0c\u5b9a\u4e49\n\\begin{align*}\n\\left(\\frac{n}{0}\\right)=\\left\\{\\begin{array}{cc}\n1&amp;\\qquad n=\\pm1\\\\\n0&amp;\\qquad \\text{\u5176\u4ed6}\n\\end{array}\\right.\n\\end{align*}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6027\u8d28<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u8bbe$K=\\Q(\\sqrt{d}),m=\\operatorname{disc}(K)$\u3002\u4e8c\u6b21\u57df\u5bf9\u5e94\u7684\u4e8c\u6b21\u72c4\u91cc\u514b\u96f7\u7279\u5f81\u6070\u7531\u514b\u7f57\u5185\u514b\u7b26\u53f7\u7ed9\u51fa\uff0c\u5373\n\\[\n\\chi_m(n)=\\left(\\frac{m}{n}\\right)\n\\]\n\u662f\u6a21$\\left| m \\right|$\u7684\u672c\u539f\u4e8c\u6b21\u72c4\u91cc\u514b\u96f7\u7279\u5f81\u3002\u5b83\u5bf9\u5e94\u4e86\u8d28\u6570\u5728$K$\u4e2d\u7684\u5206\u89e3\u884c\u4e3a\uff1a\n\\begin{align*}\n\\chi_{m}(p)=\n\\begin{cases}\n1,&amp;p\\text{ \u5728 }K\\text{ \u4e2d\u5206\u88c2},\\\\\n-1,&amp;p\\text{ \u5728 }K\\text{ \u4e2d\u60f0\u6027},\\\\\n0,&amp;p\\text{ \u5728 }K\\text{ \u4e2d\u5206\u6b67}.\n\\end{cases}\n\\end{align*}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e8e\u662f\uff0c\u6234\u5fb7\u91d1$\\zeta$\u51fd\u6570\u7684\u6b27\u62c9\u56e0\u5b50\u53ef\u5206\u89e3\u4e3a\uff1a\n\\[\n\\zeta_K(s)=\\zeta(s)L(s,\\chi_{m})\n\\]<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_import_markdown_pro_load_document_selector":0,"_import_markdown_pro_submit_text_textarea":"","footnotes":""},"categories":[31,67,3],"tags":[72,71,88],"class_list":["post-1838","post","type-post","status-publish","format-standard","hentry","category-algebra","category-algebraic-number-theory","category-mathematics","tag-algebra","tag-71","tag-88"],"_links":{"self":[{"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/posts\/1838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1838"}],"version-history":[{"count":26,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/posts\/1838\/revisions"}],"predecessor-version":[{"id":1871,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/posts\/1838\/revisions\/1871"}],"wp:attachment":[{"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/okb385.org.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}