New PDF release: Algebra und Zahlentheorie [Lecture notes]

By Walter Gubler

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Additional resources for Algebra und Zahlentheorie [Lecture notes]

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3. Sei G eine endliche Gruppe und p ein Primteiler von ord(G), dann gibt es ein g ∈ G mit ord(g) = p. 1 existiert eine p-Sylow-Untergruppe H von G. Sei h ∈ H\{c}. 8. Damit ist ord(h) = pl f¨ ur ein l ∈ {1, . . , m} l−1 und hieraus k¨onnen wir weiterhin folgern, dass g := hp = e und es l−1 l gilt g p = (hp )p = hp = e. 14). Weil p eine Primzahl ist, muss ord(g) = p gelten. 48 KAPITEL 2. 4. , Gr Gruppen. Auf G1 × · · · × Gr definieren wir eine Gruppenstruktur durch (g1 , . . , gr ) · (g1 , . .

8. Bekanntlich ist Q der kleinste K¨ orper der Z enth¨alt. Diese Uberlegung kennen wir schon aus der Schule. Bez¨ uglich der Addition ist Z ja bereits eine kommutative Gruppe. Bez¨ uglich der Multiplikation haben wir gesehen, dass Z ein Monoid ist. Um also aus Z einen K¨orper zu machen, ben¨otigen wir also noch die jeweils inversen Elemente. Dazu f¨ uhrt man Br¨ uche ein und erh¨ alt so Q. Diesen Ansatz verallgemeinern wir im Folgenden. Zun¨achst aber wollen wir noch einen Gedanken aufgreifen, den die meisten Leser bereits so verinnerlicht haben, dass man ihn vollends intuitiv gebraucht.

Gr werden genau dann Erzeugende eines Ideals I genannt, wenn I =< g1 , . . , gr > erf¨ ullt ist. 3. Ein Ideal I R heißt genau dann Hauptideal, wenn es ein g ∈ I gibt so, dass I =< g >, also I = Rg. Weiter heißt ein Ring R genau dann Hauptidealbereich, wenn R ein Integrit¨atsbereich und jedes Ideal von R ein Hauptideal ist. 4. Z ist ein Hauptidealbereich. 62 KAPITEL 3. RINGTHEORIE Beweis: Dass es sich bei Z um einen Integrit¨atsbereich handelt ist bereits bekannt. Sei nun I ein Ideal in Z. F¨ ur den endg¨ ultigen Nachweis m¨ ussen wir nun zeigen, dass es dann ein g ∈ Z gibt so, dass I = Z · g.

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Algebra und Zahlentheorie [Lecture notes] by Walter Gubler

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