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Download e-book for kindle: Algebraische Zahlentheorie [Lecture notes] by Rainer Vogt

By Rainer Vogt

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Sommersemester 2010

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Example text

Beweis: DF/K (α1 , . . , αn ) = det(SF/K (αi · αj )) = det( k σk (αi · αj )) = det( k σk (αi ) · σk (αj )) = det((σk (αi ))i,k · (σk (αj ))k,j ) = det(σk (αj ))2 . 6 Definition: Sei f ∈ K[X] ein Polynom vom Grad n und Leitkoeffizin ent c. Sei f = c · dann nennt man i=1 (X − αi ) seine Faktorisierung in Linearfaktoren in K[X], D(f ) = 1≤i

Es folgt OK /G ∼ = Z/a1 × . . × Z/an . 2 folgt n DK/Q (β1′ , . . , βn′ ) =( i=1 a2i )·DK/Q (α1′ , . . , αn′ ) = |OK /G|2 ·DK/Q (α1′ , . . , αn′ ). Die Transformationsmatrizen T1 zwischen den Basen {β1 , . . , βn } und {β1′ , . . , βn′ } und T2 zwischen den Basen {α1 , . . , αn } und {α1′ , . . , αn′ } sind ganzzahlige invertierbare Matrizen, so dass det(Ti ) ∈ Z∗ , also (det Ti )2 = 1. Es folgt DK/Q (β1 , . . , βn ) = DK/Q (β1′ , . . , βn′ ) und DK/Q (α1 , . . , αn ) = DK/Q (α1′ , . .

N−1 ) = D(f ). Beweis: Da α separabel ist, hat f verschiedene Nullstellen α1 , . . , αn in K, und wir haben n verschiedene K-Morphismen. σi : F → K, α → αi . 5  2 1 α1 α12 . . α1n−1  ..   . α2 α22 . . α2n−1  n−1 j−1 2 DF/K (1, α, . . , α ) = det(σi (α )) = det  . ..   .. .  2 n−1 1 αn αn . . αn Die rechte Seite ist das Quadrat der Vandermonde-Determinante. Aus ¨ (αi − αj ))2 . 7 gilt DF/K (1, α, . . , αn−1 ) = (−1)n·(n−1)/2 · NF/K (f ′ (α)). n Beweis: f = (X − αi ), f ′ = j=1 n NF/K (f ′ (α)) = n j=1 i=1 i=j (X − αi ).

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Algebraische Zahlentheorie [Lecture notes] by Rainer Vogt


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