\u4ee5\u4e0b\uff0c\u5168\u304f\u306e\u500b\u4eba\u7684\u30e1\u30e2\u3002<\/p>\n
Gauss\u306e\u5186\u5468\u7b49\u5206\u8ad6\u306b\u3042\u308b\u306e\u3060\u304c\uff0c\u4e00\u898b\u3059\u308b\u3068\u95a2\u4fc2\u306a\u3055\u305d\u3046\u306a\u4e0d\u601d\u8b70\u306a\u3082\u306e\u304c\uff0cCyclotomic numbers (\u5186\u5206\u6570\u3068\u3067\u3082\u8a33\u3059\u306e\u304b\uff1f)\u306a\u308b\u3082\u306e\u3002\u521d\u7b49\u7684\u8b70\u8ad6\u3067\uff0cCyclotomic numbers \u306b\u3064\u3044\u3066\u306e\u6027\u8cea\u304c\u793a\u305b\u308b\u3053\u3068\u304c\u5206\u304b\u3063\u305f\u306e\u3067\uff0c\u30e1\u30e2\u3002\u7c21\u5358\u306e\u305f\u3081\uff0c$p=4l+3$\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u306e\u307f\u3002<\/p>\n
$p$\u3092\u7d20\u6570\u3068\u3057\u3066\uff0cmodulo $p$\u3067\u306e\u5e73\u65b9\u5270\u4f59\u3092$R$\u3067\u8868\u3057\uff0c\u5e73\u65b9\u975e\u5270\u4f59\u3092$N$\u3067\u8868\u3059\u3002$1\\leqq a \\leqq p-2$\u306a\u308b$a$\u306b\u5bfe\u3057\u3066\uff0c$a\\in R$, $a+1\\in R$\u306a\u308b$a$\u306e\u500b\u6570\u3092$(RR)$\u3068\u66f8\u304f\u3002\u540c\u69d8\u306b\uff0c$a\\in N$, $a+1\\in R$\u306a\u308b$a$\u306e\u500b\u6570\u3092$(NR)$\u3068\u66f8\u304f\u3002$(RN)$, $(NN)$\u3082\u540c\u69d8\u306b\u5b9a\u7fa9\u3059\u308b\u3002\u3053\u308c\u3089\u306e\u6570\u3092\u6c42\u3081\u308b\u306e\u304c\u76ee\u6a19\u3002<\/p>\n
$p=4l+3$\u306e\u3068\u304d\u306b\u306f\uff0c$(RR)=(NN)=(NR)$\u304c\u6210\u308a\u7acb\u3064\u3002\u3053\u308c\u304c\u9375\u3068\u306a\u308b\u3002\u307e\u305a\uff0c$a\\in R$, $a+1\\in R$\u3068\u3059\u308b\u3002$p=4l+3$\u306e\u3068\u304d\u306f\uff0c$-1$\u304c\u975e\u5270\u4f59\u3060\u304b\u3089\uff0c$-a\\in N$, $-a-1\\in R$\u3068\u306a\u308b\u3002\u5bfe\u5fdc$a \\longleftrightarrow -a-1$\u306b\u3088\u3063\u3066\uff0c$(RR)=(NN)$\u3068\u306a\u308b\u3002<\/p>\n
$(NR)=(NN)$\u306e\u8a3c\u660e\u306b\u306f\uff0c$a$\u306emodulo $p$\u3067\u306e\u9006\u5143$b$\uff0c\u3064\u307e\u308a\uff0c$ab\\equiv 1 \\pmod{p}$\u306a\u308b$b$\u3092\u7528\u3044\u308b\u3002$a\\in N$, $a+1\\in R$\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\uff0c$b\\in N$\u3067\u3042\u308a\uff0c\u307e\u305f\uff0c
\n\\[ 1+b=ab+b=(a+1)b \\in N \\]
\n\u3067\u3042\u308b\u304b\u3089\uff0c$b\\in N$, $b+1\\in N$\u3068\u306a\u308b\u3002\u6545\u306b\uff0c\u5bfe\u5fdc $a\\longleftrightarrow b$\u306b\u3088\u3063\u3066\uff0c$(NR)=(NN)$\u304c\u6210\u308a\u7acb\u3064\u3002<\/p>\n
\u3042\u3068\u306f\u7c21\u5358\u3002$a\\in R$\u306e\u3068\u304d\uff0c$a+1$\u306f$R$, $N$\u306e\u3044\u305a\u308c\u304b\u3060\u304b\u3089\uff0c
\n\\[ (RR)+(RN)=\\frac{p-1}{2} \\]
\n\u304c\u6210\u308a\u7acb\u3064\u3057\uff0c\u5b9a\u7fa9\u304b\u3089\uff0c
\n\\[ (RR)+(RN)+(NR)+(NN)=p-2 \\]
\n\u306f\u660e\u3089\u304b\u3002\u4ee5\u4e0a\u304b\u3089\uff0c
\n\\[ (RR)=(NN)=(NR)=\\frac{p-3}{4}, \\quad (RN)=\\frac{p+1}{4} \\]
\n\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u4ee5\u4e0b\uff0c\u5168\u304f\u306e\u500b\u4eba\u7684\u30e1\u30e2\u3002 Gauss\u306e\u5186\u5468\u7b49\u5206\u8ad6\u306b\u3042\u308b\u306e\u3060\u304c\uff0c\u4e00\u898b\u3059\u308b\u3068\u95a2\u4fc2\u306a\u3055\u305d\u3046\u306a\u4e0d\u601d\u8b70\u306a\u3082\u306e\u304c\uff0cCyclotomic numbers (\u5186\u5206\u6570\u3068\u3067\u3082\u8a33\u3059\u306e\u304b\uff1f)\u306a\u308b\u3082\u306e\u3002\u521d\u7b49\u7684\u8b70\u8ad6\u3067\uff0cCyclotomic numb […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[37,5],"tags":[],"class_list":["post-1807","post","type-post","status-publish","format-standard","hentry","category-37","category-5"],"_links":{"self":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/1807","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/comments?post=1807"}],"version-history":[{"count":19,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/1807\/revisions"}],"predecessor-version":[{"id":1827,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/1807\/revisions\/1827"}],"wp:attachment":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/media?parent=1807"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/categories?post=1807"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/tags?post=1807"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}