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13 Definition. Sei K ein K¨orper mit char(K) = p > 0. Dann heißt ϕ : K → K, ϕ(a) = ap (a ∈ K) 3 der Frobenius (-Endomorphismus) von K. Man schreibt K p := {ap : a ∈ K}. 14 Bemerkungen. 1. Sei K ein K¨ orper mit char(K) = p > 0. Dann ist K p ein Teilk¨orper von K, und der Frobenius ist ein Isomorphismus von K auf K p . 2. Es ist ϕ(1) = 1, also ist f¨ ur a ∈ Fp auch ϕ(a) = a. Anders gesagt, es ist np ≡ n (mod p) f¨ ur alle n ∈ Z. Ist n ≡ 0 (mod p), so folgt np−1 ≡ 1 (mod p) (Kleiner Satz von Fermat) durch K¨ urzen im K¨orper Fp .

Die h¨ oheren Ableitungen von f sind induktiv definiert durch f (k) := f (k−1) (k ≥ 1). Es handelt sich wohlgemerkt um eine rein formale Definition, bei der kein Grenzprozeß beteiligt ist. 4 Regeln. F¨ ur f , g ∈ K[x] und a, b ∈ K gilt: (a) (af + bg) = af + bg (Linearit¨ at), 32 ¨ ¨ 2. KORPERTHEORIE I: ENDLICHE KORPERERWEITERUNGEN (b) (f g) = f g + f g (Produktregel), (c) (f ◦ g) = (f ◦ g) · g (Kettenregel). 5 Satz. Sei 0 = f ∈ K[x], sei a ∈ K mit f (a) = 0. Dann ist µ(f , a) ≥ µ(f, a) − 1. Dabei gilt Gleichheit genau dann, wenn char(K) kein Teiler von µ(f, a) ist.

Quadratur des Kreises: Zu einem Kreis konstruiere man ein Quadrat mit demselben Fl¨ acheninhalt. Alle drei Probleme sind mit Zirkel und Lineal unl¨osbar: 1. Sei α = eiϑ √ mit ϑ ∈ R. Der Winkel ϑ l¨aßt sich genau dann in drei gleiche Teile 3 teilen, wenn man α aus {0, 1, α} konstruieren kann. 12 ist ¨aquivalent, daß Q(α) eine dritte Wurzel aus α enth¨alt. Ist α transzendent u ¨ber Q, so ist x3 −α irreduzibel u ur algebraisches ¨ber Q(α), also die Dreiteilung unm¨oglich. Sie ist aber auch f¨ α meistens unm¨ oglich.

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Algebra [Lecture notes] by Claus Scheiderer


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