{"id":590,"date":"2009-01-12T10:47:49","date_gmt":"2009-01-12T01:47:49","guid":{"rendered":"http:\/\/njet.oops.jp\/wordpress\/2009\/01\/12\/2%e6%ac%a1%e5%bd%a2%e5%bc%8f%e3%83%a1%e3%83%a22\/"},"modified":"2011-04-10T22:59:14","modified_gmt":"2011-04-10T13:59:14","slug":"2%e6%ac%a1%e5%bd%a2%e5%bc%8f%e3%83%a1%e3%83%a22","status":"publish","type":"post","link":"https:\/\/njet.oops.jp\/wordpress\/2009\/01\/12\/2%e6%ac%a1%e5%bd%a2%e5%bc%8f%e3%83%a1%e3%83%a22\/","title":{"rendered":"2\u6b21\u5f62\u5f0f\u30e1\u30e2(2)"},"content":{"rendered":"<p><latex><br \/>\n\u30ac\u30a6\u30b9\u306e\u300c\u7a2e\u306e\u7406\u8ad6\u300d(Genus Theory) \u3078\u306e\u52d5\u6a5f\u4ed8\u3051\u3002<\/p>\n<p>\u5224\u5225\u5f0f$D=b^2-4ac=-15$\u306e2\u6b21\u5f62\u5f0f<br \/>\n\\[ (a,b,c)=ax^2+bxy+cy^2 \\]<br \/>\n\u306b\u3088\u308a\u7d20\u6570$p$\u304c\u8868\u3055\u308c\u308b\u304b\u5426\u304b\u3001\u3068\u3044\u3046\u554f\u984c\u3092\u8003\u3048\u308b\u3002<\/p>\n<p>\u3053\u306e\u5834\u5408\u3001\u7c21\u7d04\u5f62\u5f0f (reduced form) \u306f<br \/>\n\\[ (1,1,4)=x^2+xy+4y^2, \\quad (2,1,2)=2x^2+xy+2y^2 \\]<br \/>\n\u306e2\u3064\u3002\u5224\u5225\u5f0f $-15$ \u306e\u3069\u306e\u5f62\u5f0f\u3082\u3053\u306e2\u3064\u306e\u3044\u305a\u308c\u304b\u4e00\u65b9\u3068\u6b63\u5f0f\u540c\u5024\u306b\u306a\u308b\u3002<br \/>\n(\u3064\u307e\u308a\u3001$SL(2,\\mathbb{Z})$ \u3067\u4e92\u3044\u306b\u79fb\u308a\u5408\u3046\u3002)<br \/>\n\u3053\u306e2\u3064\u306f\u540c\u5024\u3058\u3083\u306a\u3044\u306e\u3067\u3001\u72ed\u7fa9\u306e\u300c\u985e\u6570\u300d(Class Number)\u306f2\u3068\u306a\u308b\u3002<\/p>\n<p>2\u6b21\u5f62\u5f0f\u306e\u4e00\u822c\u8ad6\u304a\u3088\u3073\u5e73\u65b9\u5270\u4f59\u306e\u7406\u8ad6\u306a\u3069\u304b\u3089\u3001\u6b21\u306e\u3053\u3068\u307e\u3067\u306f\u5206\u304b\u308b\u3002\u3059\u306a\u308f\u3061\u30012, 3, 5\u4ee5\u5916\u306e\u7d20\u6570$p$\u304c\u3053\u306e2\u3064\u306e\u3044\u305a\u308c\u304b\u3067\u8868\u3055\u308c\u308b\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f\u30012\u6b21\u5408\u540c\u5f0f<br \/>\n\\[ x^2 \\equiv -15 \\quad ({\\rm mod } 4p ) \\]<br \/>\n\u304c\u89e3\u3092\u3082\u3064\u3053\u3068\u3001\u3064\u307e\u308a\u3001<br \/>\n\\[ p \\equiv 1, 2, 4, 8 \\quad ({\\rm mod } 15 ) \\]<br \/>\n\u3067\u3042\u308b\u3002<br \/>\n\u3057\u304b\u3057\u3001\u3053\u308c\u3067\u306f\u3001$(1,1,4)$\u3068$(2,1,2)$\u306e\u3069\u3061\u3089\u3067\u8868\u3055\u308c\u308b\u306e\u304b\u3001\u3042\u308b\u3044\u306f\u4e21\u65b9\u3067\u8868\u3057\u3046\u308b\u306e\u304b\u3001\u304c\u5206\u304b\u3089\u306a\u3044\u3002\u305d\u306e\u70ba\u306b\u306f\u4f55\u3089\u304b\u306e\u65b9\u6cd5\u3067\u30012\u3064\u306e\u5f62\u5f0f\u3092\u300c\u5206\u96e2\u300d\u3059\u308b\u3053\u3068\u304c\u5fc5\u8981\u3068\u306a\u308b\u3002<\/p>\n<p>\u305d\u3053\u3067\u3001\u5225\u306e\u5207\u308a\u53e3\u304b\u3089\u554f\u984c\u3092\u8003\u3048\u3066\u307f\u308b\u30022\u6b21\u5f62\u5f0f\u304c\u8868\u3059\u6574\u6570\u3092\u3001\u3044\u308d\u3093\u306a\u6570\u3092\u6cd5(modulo)\u3068\u3057\u3066\u8003\u3048\u308b\u306e\u3067\u3042\u308b\u3002\u5177\u4f53\u7684\u306b\u306f\u5224\u5225\u5f0f\u3092\u5272\u308a\u5207\u308b\u7d20\u6570\u3092\u8003\u3048\u308b\u3002(\u305d\u308c\u4ee5\u5916\u306e\u5947\u7d20\u6570\u3092\u30e2\u30b8\u30e5\u30ed\u3068\u3057\u3066\u3082\u3001\u60c5\u5831\u306f\u5f97\u3089\u308c\u306a\u3044\u3053\u3068\u304c\u793a\u3055\u308c\u308b\u3002Cahen\u306e393\u30da\u30fc\u30b8\u3042\u305f\u308a\u3002)<\/p>\n<p>modulo 3 \u3067\u8003\u3048\u3066\u307f\u3088\u3046\u30023\u306e\u500d\u6570\u306b\u306a\u308b\u3082\u306e\u306f\u8003\u3048\u306a\u3044\u3053\u3068\u306b\u3057\u3066\u3001\u3053\u308c\u4ee5\u5916\u304c\u4f59\u308a1\u30012\u306e\u3044\u305a\u308c\u306b\u306a\u308b\u306e\u304b\u3092\u8abf\u3079\u3066\u307f\u308b\u3068\u3001<br \/>\n\\[ x^2+xy+4y^2 \\equiv -2(x-y)^2 \\equiv 1 \\quad ({\\rm mod } 3) \\]<br \/>\n\u304a\u3088\u3073<br \/>\n\\[ 2x^2+xy+2y^2 \\equiv 2(x+y)^2 \\equiv 2 \\quad ({\\rm mod } 3) \\]<br \/>\n\u3068\u306a\u308b\u3002<\/p>\n<p>modulo 5 \u3067\u3082\u540c\u69d8\u306e\u7d50\u679c\u306b\u306a\u308b\u30025\u306e\u500d\u6570\u306b\u306a\u308b\u3082\u306e\u3092\u8003\u3048\u306a\u3044\u3053\u3068\u306b\u3059\u308c\u3070\u3001<br \/>\n\\[ x^2+xy+4y^2 \\equiv -(2x-y)^2 \\equiv 1, 4 \\quad ({\\rm mod } 5) \\]<br \/>\n\\[ 2x^2+xy+2y^2 \\equiv 2(x-y)^2 \\equiv 2, 3 \\quad ({\\rm mod } 5) \\]<br \/>\n\u3067\u3042\u308b\u3002<\/p>\n<p>\u3053\u306e\u554f\u984c\u306e\u5834\u5408\u306f modulo 3, modulo 5 \u306e\u7247\u65b9\u3060\u3051\u304b\u3089\u3082\u5206\u96e2\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001<br \/>\n$p\\neq 2, 3, 5$ \u306a\u308b\u7d20\u6570$p$\u306b\u5bfe\u3057\u3066\u3001<br \/>\n\\[ p=x^2+xy+4y^2 \\quad \\Longleftrightarrow\\quad p \\equiv 1, 4 \\quad ({\\rm mod } 15) \\]<br \/>\n\u304a\u3088\u3073\u3001<br \/>\n\\[ p=2x^2+xy+2y^2 \\quad \\Longleftrightarrow\\quad p \\equiv 2, 8 \\quad ({\\rm mod } 15) \\]<br \/>\n\u3068\u3044\u3046\u7d50\u8ad6\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n<p>\u4ee5\u4e0a\u3092\u4e00\u822c\u5316\u3059\u308b\u306b\u306f\u3001\u5e73\u65b9\u5270\u4f59\u306e\u6307\u6a19(character)\u3068\u95a2\u9023\u3065\u3051\u308b\u3002modulo 3 \u3067\u7279\u5b9a\u306e\u4f59\u308a\u306b\u306a\u308b\u3053\u3068\u306f\u3001\u6307\u6a19 $\\left(\\frac{n}{3}\\right)$ (\u30eb\u30b8\u30e3\u30f3\u30c9\u30eb\u306e\u8a18\u53f7\u3001\u30e4\u30b3\u30d3\u306e\u8a18\u53f7) \u304c\u4e00\u5b9a\u306e\u5024\u306b\u306a\u308b\u3053\u3068\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u308b\u3002\u5224\u5225\u5f0f\u3092\u5272\u308a\u5207\u308b\u5947\u7d20\u6570$p_i$\u306b\u95a2\u3059\u308b\u6307\u6a19 $\\left(\\frac{n}{p_i}\\right)$ \u306e\u5024\u3092\u8003\u3048\u3001\u3053\u308c\u3089\u304c\u3059\u3079\u3066\u4e00\u81f4\u3059\u308b2\u6b21\u5f62\u5f0f\u306e\u985e(class)\u3092\u307e\u3068\u3081\u3066\u3001\u7a2e(genus)\u3068\u540d\u4ed8\u3051\u308b\u306e\u3067\u3042\u308b\u3002<br \/>\n(\u3053\u308c\u4ee5\u5916\u306b2\u306e\u30d9\u30ad\u4e57\u3067\u306e\u5206\u985e\u3001\u3064\u307e\u308a\u3001\u7d20\u65702\u306b\u95a2\u3059\u308b\u6307\u6a19\u3082\u3042\u308b\u304c\u3001\u307e\u305f\u5225\u306e\u6a5f\u4f1a\u306b\u3002)<br \/>\n<\/latex><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u30ac\u30a6\u30b9\u306e\u300c\u7a2e\u306e\u7406\u8ad6\u300d(Genus Theory) \u3078\u306e\u52d5\u6a5f\u4ed8\u3051\u3002 \u5224\u5225\u5f0f$D=b^2-4ac=-15$\u306e2\u6b21\u5f62\u5f0f \\[ (a,b,c)=ax^2+bxy+cy^2 \\] \u306b\u3088\u308a\u7d20\u6570$p$\u304c\u8868\u3055\u308c\u308b\u304b\u5426\u304b\u3001\u3068\u3044\u3046\u554f\u984c\u3092\u8003\u3048 [&hellip;]<\/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-590","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\/590","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=590"}],"version-history":[{"count":1,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/590\/revisions"}],"predecessor-version":[{"id":1738,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/posts\/590\/revisions\/1738"}],"wp:attachment":[{"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/media?parent=590"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/categories?post=590"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/njet.oops.jp\/wordpress\/wp-json\/wp\/v2\/tags?post=590"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}