By Christoph Schweigert

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Extra resources for Algebra II [Lecture notes]

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2m1 (m2 −1) ) , also p das Produkt zweier nat¨ urlicher Zahlen gr¨oßer als Eins ist. 4. F¨ ur k ∈ N ∪ {0} heißt die Zahl k Fk = 22 + 1 k-te Fermatzahl. Die Zahlen F0 = 3 F1 = 5 F2 = 17 F3 = 257 F4 = 65537 sind alle Primzahlen. Fermat selbst hat behauptet, alle Fermatzahlen Fk seien prim, aber die Fermatzahlen Fk mit 5 ≤ k ≤ 16 sind keine Primzahlen. Es ist zur Zeit keine weitere Fermat-Primzahl Fk mit k > 4 bekannt. 2 Das quadratische Reziprozit¨ atsgesetz Als Motivation schildern wir die folgende Fragestellung: sei p = 2 eine Primzahl und E = Q(ζp ) der p-te Kreisteilungsk¨orper.

3 (ii) zyklisch von der Ordnung pr − 1, also hat die × Untergruppe F×2 pr der Elemente von Fpr , die sich als Quadrat schreiben lassen, Index zwei: ×2 [F× ur pr = 2 . 5. Sei ν ∈ Z, ν = 0. Dann setze ν p := +1 falls ν = x2 mod p f¨ ur ein x ∈ Z 2 −1 falls ν = x mod p f¨ ur alle x ∈ Z ν In Worten: νp = 1 genau dann, wenn ν mod p ein Quadrat in F× p ist. p heißt das Legendre-Symbol und h¨angt nur von der Restklasse von ν modulo p ab. ×2 F¨ ur p = 2 zeigt (10), dass die Faktorgruppe F× pr /Fpr isomorph zu Z2 ist.

Iv) Sei ζn eine primitive n-te Einheitswurzel und σ ∈ G(Q(ζn )/Q). Es gilt Fnσ (X) = (X − σ(ζ)) = Fn (X) . ord ζ=n Da die Erweiterung Q(ζn )/Q galoisch ist, heißt dies, dass alle Koeffizienten von Fn im Fixk¨orper Q liegen, Fnσ ∈ Q[X]. Da außerdem Fn (X) das Polynom X n − 1 teilt, gibt es ein Polynom g ∈ Q[X] mit Fn (X)g(X) = X n − 1. Nach dem Lemma von Gauß folgt Fn (X) ∈ Z[X]. 16. Das Kreisteilungspolynom Fn (X) ist irreduzibel im Polynomring Q[X] u ¨ber den rationalen Zahlen. Beweis. Sei ζn eine primitive n-te Einheitswurzel.

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Algebra II [Lecture notes] by Christoph Schweigert
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