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Extra info for Algebraische Zahlentheorie [Lecture notes]

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3 Beispiel: Z · 01 + Z · 10 ⊂ R2 ist das Gitter, das aus allen Punkten in Z × Z ⊂ R × R besteht. Das Einheitsquadrat ist eine Grundmasche. 59 111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 Dieses Gitter ist vollst¨andig. ∈ Z2 aber auch in der Form Da sich m n m n 11111111111 00000000000 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 = (m − n) 1 1 +n· 0 1 darstellen l¨aßt, ist dieses Gitter auch durch Z · 10 + Z · 11 mit nebenstehender Grundmasche gegeben.

A0 aus K[X] das Minimalpolynom von α ∈ F . Dann gilt mit m = [F : K(α)] (1) PK(α)/K (α; X) = f (X) (2) SK(α)/K (α) = −an−1 (3) NK(α)/K (α) = (−1)n a0 34 (4) PF/K (α; X) = f (X)m (5) SF/K (α) = −man−1 (6) NF/K (α) = (−1)m·n · am 0 Beweis: {1, α, α2 , . . 3 eine K-Basis f¨ ur K(α). Bzgl. dieser Basis hat αK(α)/K : K(α) → K(α) die Matrix   −a0 −a 1   ,    @ 1 −an−1 0@ 1@@@  @ @ C(α) =  @ 0  0 PK(α)/K (α) = det(X · En − C(α)) = f Damit ist (1) gezeigt, und (2), (3) folgen. 6 auf K ⊂ K(α) ⊂ F an.

N ) = µ2 · DA/R (α1 , . . , αn ) Beweis: Sei βi = mit µ ∈ R∗ . rij · αj mit rij ∈ R. Dann gilt DA/R (β1 , . . , βn ) = det(rij )2 · DA/R (α1 , . . , αn ) {β1 , . . , βn } ist genau dann R-Basis von A, wenn det(rij ) invertierbar ist. Da R Integrit¨atsring ist, entspricht det(rij )2 dem µ2 . Wieder interessiert uns besonders der Fall von endlichen K¨orpererweiterungen F/K. 5 Satz: Ist F/K endliche separable K¨orpererweiterung von Grad n und sind σi : F → K die n verschiedenen K-Morphismen, dann gilt f¨ ur α1 , .

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