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Algebra I by Walter Gubler PDF

By Walter Gubler

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M. 2. KORPERERWEITERUNGEN 65 Im zweiten Schritt zeigen wir die Behauptung, falls m := [M : L] < ∞ und l := [L : K] < ∞. Wir w¨ ahlen eine Basis β1 , . . , βl von L als K-Vektorraum und eine Basis γ1 , . . , γm von M als L-Vektorraum. Nach dem ersten Schritt wissen wir, dass (βi γj )1≤i≤l,1≤j≤m K-linear unabh¨angig in M ist. Um nun das gew¨ unschte [M : K] = ml zu zeigen, gen¨ ugt es zu beweisen, dass (βi γj )1≤i≤l,1≤j≤m ein K-Erzeugendensystem in M bildet (weil wir damit eine K-Basis erhalten).

8 gibt es x, y ∈ R mit 1 = ax + by und damit =⇒ c = cax + cby =⇒ a | c. 10. Sei R ein Hauptidealbereich, I ein Primideal =⇒ I = {0} oder I ist ein Maximalideal. Beweis. Siehe Aufgabe 28. 11. Z hat die Ideale nZ, n ≥ 0. 5 alle verschieden und es gilt: Maximalideale ⇐⇒ pZ, p prim. 12 (Chinesischer Restsatz f¨ ur einen Hauptidealbereich R). Seien g1 , . . , gr paarweise teilerfremd in R\ {0}. Dann gilt: ∼ R/ g1 · . . · gr −→ (R/ g1 ) × . . × (R/ gr ) Beweis. 5 gk + gl = R ⇐⇒ ggT(gk , gl ) ∼ 1. 15.

Wir betrachten also in diesem Abschnitt ein kommutatives Monoid M , das die K¨ urzungsregel erf¨ ullt. Wir schreiben die Verkn¨ upfung von M multiplikativ. Dann bedeutet die K¨ urzungsregel folgendes: ab = ac ⇒ b = c ∀a, b, c ∈ M. In den Anwendungen wird M = R\ {0} sein f¨ ur einen Integrit¨atsbereich R. 6. 1. a, b ∈ M heißen assoziiert :⇐⇒ a | b und b | a. Wir schreiben dann a ∼ b. ¨ Weil | transitiv ist, muss ∼ eine Aquivalenzrelation sein. Aus der K¨ urzungsregel folgt sofort a ∼ b ⇐⇒ ∃u ∈ M ∗ (Einheit) mit a = ub.

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