By Christoph Schweigert
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Additional info for Algebra II [Lecture notes]
Example text
F¨ ur p = 2 gilt σ √ √ p∗ = χ(σ) p∗ f¨ ur alle σ ∈ G(Q(ζ)/Q). 41 Beweis. Sei σ = σa mit a koprim zu p, so gilt: σ(S) = σa (S) = ν = a p ν aν ζ = p ν νa−1 ν ζ p ν ν ζ = χ(σ)S . 10 (quadratisches Reziprozit¨atsgesetz von Gauß). (i) Seien p, q zwei verschiedene Primzahlen ungleich zwei, p, q = 2 und p = q. Dann gilt p−1 q−1 q p = (−1) 2 2 p q In Worten: Gilt p = 1 mod 4 oder q = 1 mod 4, so ist q ein quadratischer Rest modulo p genau dann, wenn p ein quadratischer Rest modulo q ist. Gilt p = q = 3 mod 4, so ist q quadratischer Rest modulo p genau dann, wenn p nicht quadratischer Rest q ist.
Die Abbildung E×E → K (α, β) → SpE/K (αβ) ist eine symmetrische K-Bilinearform auf E. (ii) Die Norm NE/K : E × → K × ist ein Gruppenhomomorphismus auf den Einheitengruppen. (iii) Ist PE/K (α) = X n + an−1 X n−1 + · · · + a0 ∈ K[X] das charakteristische Polynom, so gilt SpE/K (α) = −an−1 NE/K (α) = (−1)n a0 . 3. 61 (i) Sei F ein Zwischenk¨orper von E/K und m := [E : F ]. Dann gilt f¨ ur α ∈ F: PE/K (α) = PF/K (α)m SpE/K (α) = m SpF/K (α) NE/K (α) = NF/K (α)m (ii) Sei E/K eine endliche Erweiterung, α ∈ E und f = minK (α) = X n + .
Keine endliche Erweiterung. 1. Sei p eine Primzahl, K = Fp . Da K perfekt ist, gilt C = Cs . Wir bezeichnen diesen K¨orper mit Fp∞ . Mit ϕ ∈ G(Fp∞ /Fp ) bezeichnen wir den Frobeniusautomorphismus ϕ : Fp∞ → Fp∞ x → xp Sei H = ϕ die vom Frobeniusautomorphismus zyklisch erzeugte Untergruppe der Galoisgruppe G(Fp∞ /Fp ). Der zugeh¨orige Fixk¨orper ist FH p∞ = Fp . 50 W¨ urde der Hauptsatz der Galoistheorie so stimmen, wie er f¨ ur endliche Erweiterungen formuliert wurde, so w¨ urde daraus f¨ ur die Galoisgruppe folgen H = G(Fp∞ /Fp ) .
Algebra II [Lecture notes] by Christoph Schweigert
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