Algebraische Zahlentheorie [Lecture notes] by J. Sander et al.

By J. Sander et al.

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Es gibt h(x) ∈ Z[x] mit m2 (xp ) = m1 (x) · h(x) . Wir betrachten die Polynomidentit¨aten u ¨ber Z nun modulo p. Man zeigt leicht mit dem Binomischen Lehrsatz, dass f¨ ur beliebige Polynome f (x) gilt f (xp ) ≡ f (x)p mod p. Also haben wir m2 (x)p ≡ m1 (x) · h(x) mod p . Ist nun k(x) ein beliebiger irreduzibler Faktor von m1 (x) mod p, so ist k(x) auch ein solcher Teiler von m2 (x)p und damit von m2 (x) mod p. h. k(x) | (xn − 1) mod p und k(x) | (xn − 1) = nxn−1 mod p. Wegen p n hat n · xn−1 nur irreduzible Faktoren x, die jedoch nicht xn − 1 teilen.

1            hat die Determinante det A = r = 0. 38 ist auch B2 = {δ, β2 , β3 , . . , βd } ⊆ OF eine Q-Basis von F . 35 haben wir discr (B2 ) = r2 · discr (B1 ), also |discr (B2 )| < |discr (B1 )| . h. als Z-Modul haben wir OF = Zβ1 ⊕ · · · ⊕ Zβd . 40 Ist B ⊆ OF eine Basis von F u ¨ber Q und ist discr (B) quadratfrei, so ist B eine Ganzheitsbasis von F . Beweis: Sei B = {β1 , β2 , . . , βd } ⊆ OF . 39 besitzt F eine Ganzheitsbasis B1 = {α1 , α2 , . . , αd }. 35 gilt discr (B) = D2 · discr (B1 ) , wobei D = det(qi,k ) mit qi,k ∈ Z definiert durch d (∗) βk = qi,k αi i=1 (k = 1, .

Sei d δ := γ − [q1 ]β1 = j=1 d qj βj − [q1 ]β1 = rβ1 + j=2 qj βj ∈ OF . 1 ALGEBRAISCHE ZAHLEN Die Matrix       A=     35 r q2 q3 · · · qd 0 ... 1 0 ··· 0 .. 1 0 0 .. ··· ··· 0 ... 1            hat die Determinante det A = r = 0. 38 ist auch B2 = {δ, β2 , β3 , . . , βd } ⊆ OF eine Q-Basis von F . 35 haben wir discr (B2 ) = r2 · discr (B1 ), also |discr (B2 )| < |discr (B1 )| . h. als Z-Modul haben wir OF = Zβ1 ⊕ · · · ⊕ Zβd . 40 Ist B ⊆ OF eine Basis von F u ¨ber Q und ist discr (B) quadratfrei, so ist B eine Ganzheitsbasis von F .

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