3. 今⽇のテーマ
p = x2 + ny2 の形で表される素数の法則
3
n を正の整数とする
「整数 x, y が存在して p = x2 + ny2」が成り⽴つ素数 p の条件は?
「整数 x, y が存在して p = x2 + ny2」が成り⽴つとき
「x2 + ny2 は p を表現する」という
48. ヒルベルトの分岐理論の設定
48
K (代数体)
L (Kのガロア拡⼤体)
K の素イデアル p
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ガロア拡⼤
Pe
1Pe
2 · · · Pe
g
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49. ヒルベルトの分岐理論の設定
49
K (代数体)
L (Kのガロア拡⼤体)
K の素イデアル p
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e = 1 のとき 不分岐
g = [L : K] 完全分解
ガロア拡⼤
Pe
1Pe
2 · · · Pe
g
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50. ヒルベルトの分岐理論の設定
50
K (代数体)
L (Kのガロア拡⼤体)
K の素イデアル p
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K の素イデアル が L で素イデアル分解する様⼦を調べたいp
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ガロア拡⼤
Pe
1Pe
2 · · · Pe
g
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e = 1 のとき 不分岐
g = [L : K] 完全分解