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ピタゴラス数とヒルベルトの定理90
日曜数学者 tsujimotter
2021.08.03 #3分で数学を語る会
原始ピタゴラス数
c
b
a
a2 + b2 = c2
a, b, c 整数、互いに素
(a, b, c) = (m2 ‒ n2, 2mn, m2 + n2)
m, n は整数
2
これの面白い求め方!!
3
<latexit sha1_base64="AxdC4eMIyqh6L+v1XDhtbwYS8P0=">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</latexit>
Q(
p
1)
ガウス数体
<latexit sha1_base64="qi/9WflKTc8nMUQfscNb/L10iog=">AAAGRHicpVTLbtNAFL0tBEp4NIUNEhuLEISAVpOWd4UUlU139N1Kdahsd5pa9QvbSRMs/wA/wIIVSEggPgMJsWKHoJ+AYFcQG4Q4cx3apiVhgS3P3Lmvc++ZGZuBY0exEFt9/YcO544cHTiWP37i5KnBwtDphcivh5act3zHD5dMI5KO7cn52I4duRSE0nBNRy6aG/eUfbEhw8j2vbm4Fciqa9Q8e822jBiqlcJEU7urGVdMPXoYxslwOdX0cX1c0x3fq7lGEMV+pmg+SPTIhipV/tqwthuxUiiKEcGPdlAot4ViRRA/U/5Qf510WiWfLKqTS5I8iiE7ZFCEd5nKJCiArkoJdCEkm+2SUsojtg4vCQ8D2g2MNayW21oPa5Uz4mgLKA6+EJEalcQH8Upsi3fitfgsfiJXqUu2BF+dEcMemAljqZpbmM0MQwYrg4/Pzv74Z5SLOab13aievcW0Rre4Jxs9BqxR3VpZfOPRk+3ZOzOl5KJ4Lr6gz2diS7xBp17jm/ViWs487VGPiVpUl93x79McjdJVnstcxRpY9fbUkOc9ldDrNIuuDPgkkG1oXLw+W/OoVYNWxTUxKo/OTMpPdViDLUNN4aNqUrthMQ+ZvwacSUbOciqPaAdj7/u/GPmdLDY8M6a6n5yYc7dok7FaPZlNMIaQAzCmzniz52mLkNvlG9HCt9r29WDb5K5ctnqwKF4bXEfKcghrzOdNYrd94CR0G7dMWS/wLnmY03ZGlWUDnspfMRJzXNixxy5XrSrymRcHbwXZSjv861ypQt7s8OiMtfishFyj4t/kVbpvrVY2WFjnLv5k7d7/ZeibHTjL2OUqxuq+GrNzWNmTVQMvY2Dneo8ui7gFKc+jf2XN55sNzvB3LO//Fx4UFkZHyjdGxqavFSsT2W+SBugcnadLwLkJxEmaonn0+JLe00f6lHub+5rbzn3PXPv72jFnqOPJ/foNqahQsg==</latexit>
x = a + b
p
1 7 ! x = a b
p
1
複素共役
<latexit sha1_base64="Egi6fCBZnUDXniB9HCi7MITwmAU=">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</latexit>
N(x) = xx = a2
+ b2
ノルム
4
<latexit sha1_base64="XwYmLXirQV2a3B0x8CfoiDpZKf8=">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</latexit>
= {a + b
p
1 | a, b 2 Q}
<latexit sha1_base64="ea+7egHMJ2tG1GJshU54sBAvibw=">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</latexit>
N(x) = 1 =) ある y が存在して
<latexit sha1_base64="1mIkNsZ/bSwgvqkdrQ71kJb6AsI=">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</latexit>
x =
y
y
ノルムが1なら こうかける
ここで、ヒルベルトの定理90を使うと
5
<latexit sha1_base64="ea+7egHMJ2tG1GJshU54sBAvibw=">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</latexit>
N(x) = 1 =) ある y が存在して
<latexit sha1_base64="1mIkNsZ/bSwgvqkdrQ71kJb6AsI=">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</latexit>
x =
y
y
ノルムが1なら こうかける
単位円の有理点
<latexit sha1_base64="9gi8ybVFCQTUK2A8uQKGOG/6gEo=">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</latexit>
a2
+ b2
= 1
(a, b)
6
<latexit sha1_base64="ea+7egHMJ2tG1GJshU54sBAvibw=">AAAGInicpVS7bhNBFL0JGIJ5JIEGiWaFMQoIReOEN0KyoEmBIO9Eiq1odzNxVtmXZtcvVm4p+AEKKhAUCFr4ARpEhxBFJH4AUQaJhoIzd00SO9gU7Gh37tzXuffMzFqh60SxEFsDgwcOZg4dHjqSPXrs+InhkdGTi1FQVbZcsAM3UMuWGUnX8eVC7MSuXA6VND3LlUvW5l1tX6pJFTmBPx83Q1n2zIrvrDu2GUO1OmLcH2tcMG4bhdKtdBile4FfUU5lIzaVCuqrIzkxLvgx9guFtpArCuJnOhgdrFKJ1iggm6rkkSSfYsgumRRhrFCBBIXQlSmBTkFy2C6pRVnEVuEl4WFCu4lvBauVttbHWueMONoGiotXIdKgvPgiXolt8UG8Ft/EL+TK98iW4K0youqDmTCWrrmJ2UoxZLg6/Pj03M9/RnmYY9rYjerbW0zrdJ17ctBjyBrdrZ3G1x4+2Z67OZtPzovn4jv6fCa2xHt06td+2C9n5OzTPvVYqEV32Rv/Ac3TBF3iucBVrINVf08NWd5TCX2J5tCVCZ8EsgONhxGwNYtaDWh1XANf7dGZSfvpDiuwpagt+Oia9G7YzEPqbwBnipHTnNoj2sHYO/4XI7uTxYFnylTvkxNz7ibVGavZl9kEXwU5BGP6jDf6nrYIuT2+EU28a21fH7Y6d+Wx1YdF81rjOlosK1hjPm8Sux0AJ6EbuGXaeo53ycfcamfUWTbhqf01IzHHqY499rhqXVHAvLgYRWTL7/Bf4ko1cr3DozPW5rOiuEbNv8WrVtdarxywsMFd/Mnau/+L0Dc6cFawy2V8y101puewuCerAV4mwc6VPl3mcAtaPE/8lbWAbzY4w9+x0P0v3C8sTowXro5PzlzOFe+kv0kaojN0lsaAcw2IUzRNC+jxEb2ht/Qu8yLzMfMp8zl1HRxox5yijifz9Te7JkOV</latexit>
N(x) = 1 =) ある y が存在して
<latexit sha1_base64="1mIkNsZ/bSwgvqkdrQ71kJb6AsI=">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</latexit>
x =
y
y
ノルムが1なら こうかける
単位円の有理点
<latexit sha1_base64="EbFq0jDwXOA7nd4pHgwsdObPBTQ=">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</latexit>
x = a + bi =
m + ni
m ni
<latexit sha1_base64="9gi8ybVFCQTUK2A8uQKGOG/6gEo=">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</latexit>
a2
+ b2
= 1
y = m + ni とおくと
(a, b)
7
<latexit sha1_base64="ea+7egHMJ2tG1GJshU54sBAvibw=">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</latexit>
N(x) = 1 =) ある y が存在して
<latexit sha1_base64="1mIkNsZ/bSwgvqkdrQ71kJb6AsI=">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</latexit>
x =
y
y
ノルムが1なら こうかける
単位円の有理点
<latexit sha1_base64="EbFq0jDwXOA7nd4pHgwsdObPBTQ=">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</latexit>
x = a + bi =
m + ni
m ni
<latexit sha1_base64="9gi8ybVFCQTUK2A8uQKGOG/6gEo=">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</latexit>
a2
+ b2
= 1
y = m + ni とおくと
<latexit sha1_base64="fVmbDP2vSN9fEvj/j35ZHJQVExM=">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</latexit>
=
m2
n2
m2 + n2
+
2mn
m2 + n2
i
(a, b)
8
<latexit sha1_base64="ea+7egHMJ2tG1GJshU54sBAvibw=">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</latexit>
N(x) = 1 =) ある y が存在して
<latexit sha1_base64="1mIkNsZ/bSwgvqkdrQ71kJb6AsI=">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</latexit>
x =
y
y
ノルムが1なら こうかける
単位円の有理点
<latexit sha1_base64="EbFq0jDwXOA7nd4pHgwsdObPBTQ=">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</latexit>
x = a + bi =
m + ni
m ni
<latexit sha1_base64="9gi8ybVFCQTUK2A8uQKGOG/6gEo=">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</latexit>
a2
+ b2
= 1
y = m + ni とおくと
<latexit sha1_base64="fVmbDP2vSN9fEvj/j35ZHJQVExM=">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</latexit>
=
m2
n2
m2 + n2
+
2mn
m2 + n2
i
(a, b)
a b 9
<latexit sha1_base64="ea+7egHMJ2tG1GJshU54sBAvibw=">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</latexit>
N(x) = 1 =) ある y が存在して
<latexit sha1_base64="1mIkNsZ/bSwgvqkdrQ71kJb6AsI=">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</latexit>
x =
y
y
ノルムが1なら こうかける
単位円の有理点
<latexit sha1_base64="EbFq0jDwXOA7nd4pHgwsdObPBTQ=">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</latexit>
x = a + bi =
m + ni
m ni
<latexit sha1_base64="9gi8ybVFCQTUK2A8uQKGOG/6gEo=">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</latexit>
a2
+ b2
= 1
y = m + ni とおくと
<latexit sha1_base64="fVmbDP2vSN9fEvj/j35ZHJQVExM=">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</latexit>
=
m2
n2
m2 + n2
+
2mn
m2 + n2
i
(a, b)
a b
求まった!!! 10
これ何???
11
これ何???
xは1次の
コサイクル
xは1次の
コバウンダリ
12
ヒルベルトの定理90
1次のコサイクル 1次のコバウンダリ
1次の群コホモロジー = = 0
1次のコバウンダリ
1次のコサイクル
13
ヒルベルトの定理90
1次のコサイクル 1次のコバウンダリ
1次の群コホモロジー = = 0
1次のコバウンダリ
1次のコサイクル
言い換え
14

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