Algebra [Lecture notes] by Claus Scheiderer
By Claus Scheiderer
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P i i=1 n Beweis. Sei n = − p1i ). r i=1 (1 r i=1 pei i . 13 Korollar. (Verallgemeinerter Kleiner Satz von Fermat) Sind a ∈ Z und n ∈ N mit ggT(a, n) = 1, so ist aϕ(n) ≡ 1 (mod n). Beweis. F¨ ur a = a + nZ gilt a ∈ (Z/n)∗ , also folgt aϕ(n) = 1. Der Kleine Satz von Fermat ist der Spezialfall n = p = Primzahl: F¨ ur a ∈ Z mit p a ist ap−1 ≡ 1 (mod p). 2. 2. ABELSCHE GRUPPEN 55 Zur¨ uck zur Bestimmung der Struktur von (Z/n)∗ . 14 Lemma. Sei p prim, seien x, y ∈ Z mit x ≡ y ≡ 0 (mod p). Dann ist i vp y p − x p i = vp (y − x) + i f¨ ur alle i ≥ 0, außer (eventuell) f¨ ur p = 2 und v2 (x − y) = 1.
Seien N , H Gruppen, und sei α : H → Aut(N ), h → αh ein Homomorphismus. Das semidirekte Produkt N die Menge N α H := N × H, α H von N mit H (via α) ist versehen mit der Multiplikation (n, h) · (n , h ) := n · αh (n ), hh f¨ ur (n, h), (n , h ) ∈ N × H. Dies ist eine Gruppe mit neutralem Element (e, e) und Inversen (n, h)−1 = αh−1 (n)−1 , h−1 . In N α H ist N × {e} ein zu N isomorpher Normalteiler und {e} × H ein zu H isomorphes Komplement zu diesem, und es ist (n, e)(e, h) = (n, h).
Dann ist MinPol(α/K) ein Teiler von xp − c = (x − α)p . Also ist α inseparabel u ¨ber K. 14 Satz. Sei L/K eine endliche K¨ orpererweiterung. Genau dann ist L/K rein inseparabel, wenn [L : K]s = 1 ist. Beweis. Sei [L : K]s = 1, sei α ∈ L K. 5). Umgekehrt sei L/K rein inseparabel, und sei r α ∈ L. 13 ist αp =: c ∈ K f¨ ur ein r ≥ 0 (mit p := char(K) > 0). F¨ ur jede r K-Einbettung ψ : L → K muß ψ(α)p = c sein, wodurch ψ(α) eindeutig festgelegt ist. Somit gibt es nur (h¨ ochstens) ein solches ψ, es ist also [L : K]s = 1.