Read e-book online Algebra II [Lecture notes] PDF

By Christoph Schweigert

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Extra resources for Algebra II [Lecture notes]

Example text

M ) schreiben l¨asst, wobei ωi2 ∈ K(ω1 , . . , ωi−1 ) =: Ki−1 liegt, aber ωi ∈ K(ωi , . . , ωi−1 ) liegt. Sei Em die normale H¨ ulle von Km . Dann ist die K¨orpererweiterung Em /K galoisch: normal ist sie ohnehin, und separabel ist sie, da wir in Charakteristik Null arbeiten. 36 Zu zeigen ist, dass [Em : K] Zweierpotenz ist. Wir f¨ uhren den Beweis durch Induktion nach m. F¨ ur den Induktionsanfang m = 1 betrachten wir die quadratische Erweiterung K1 = K(ω1 ). Quadratische Erweiterungen sind stets normal.

4. Schritt: Sei H ⊆ G(E/K) eine offene Untergruppe und F = E H der Fixk¨orper. Dann ist die K¨orpererweiterung F/K endlich. 8 (ii) gibt es eine endliche galoische Erweiterung L/K mit 58 L ⊆ E, so dass G(E/L) ⊆ H gilt. Damit gilt aber auch die Inklusion F = E H ⊆ E G(E/L) = L, also ist auch die K¨orpererweiterung F/K endlich. 5. Schritt: Sei H ⊆ G(E/K) eine abgeschlossene Untergruppe. Dann ist H = G(E/E H ). Beweis des fu ¨ nften Schritts: Sei H eine nicht notwendigerweise abgeschlossene Untergruppe von G(E/K).

F¨ ur m = 1 gilt |(Z/p)× | = |IF× p | = p − 1. 28 F¨ ur den Induktionsschritt betrachte wir die kanonische Surjektion Z/pm+1 Z/pm x mod pm+1 → x mod pm Sie induziert einen Gruppenhomomorphismus (Z/pm+1 )× → (Z/pm )× λ: der auch surjektiv ist. Denn aus x mod pm ∈ (Z/pm )× folgt, dass (x, pm ) = 1, und da p prim ist, folgt daraus auch, dass (x, pm+1 ) = 1 gilt. Also ist x mod pm+1 ∈ (Z/pm+1 )× Der Kern der Abbildung λ ist ker(λ) = {x mod pm+1 |x ≡ 1 mod pm } = {x = 1 + pm y mod pm+1 |y ∈ N} . Er hat offensichtlich genau p Elemente.

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