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Extra resources for Algebra und Zahlentheorie [Lecture notes]

Sample text

Wir m¨ ussen (x y)−1 xy ∈ N beweisen. 5). Analog zeigt man yx ∼ yx . 2) wohldefiniert. Da man repr¨asentantenweise vorgeht, vererben sich die Gruppenaxiome von G auf G/N . 4. G/N heißt Faktorgruppe . 5. Es gibt eine kanonische Projektion mit π : G → G/N, x → x ¯ := π(x) := xN = [x] Da wir in G/N repr¨ asentantenweise rechnen, folgt unmittelbar, dass π ein surjektiver Gruppenhomomorphismus ist. Weiterhin ist ker(π) = N . Dies muß allerdings noch separat bewiesen werden. Beweis: g ∈ ker(π) ⇔ gN = g¯ = e¯ = eN = N .

RINGTHEORIE w¨ahlen wir x ∈ ker(ϕ) und zeigen, dass a · x ∈ ker(ϕ), woraus unmittelbar folgt, dass es sich bei ker(ϕ) um ein Ideal handelt. Es gilt nun: ϕ(a · x) = ϕ(a) · ϕ(x) = ϕ(a) · 0 = 0 Damit ist also a · x ∈ ker(ϕ) und wir sind fertig. Zu bemerken ist noch, dass es zu jedem I von R einen Homomorphismus gibt, dessen Kern gleich I ist. 5 w¨ahlen. 7 (Homomorphiesatz). Sei ϕ : R1 → R2 ein Homomorphismus zwischen kommutativen Ringen, so gibt es genau einen Ringisomorphismus ϕ¯ : R1 / ker(ϕ)→ϕ(R ˜ ¯ = ϕ(x).

2. 4. Nun wollen wir diese Produktnotation auf beliebige Teilmengen von G ausdehnen. Seien hierzu also Y, Z ⊆ G gegeben. Wir setzen Y · Z := {y · z | y ∈ Y und z ∈ Z} Dies definiert eine Verkn¨ upfung auf der Potenzmenge Pot(G) von G. Die Frage ist, ob sie damit auch zu einer Gruppe wird. Die Assoziativit¨at vererbt sich trivialerweise von G und ist damit erf¨ ullt. Auch ein neutrales Element ist mit {e} gegeben, da {e}Y = {ye | y ∈ Y } = Y = Y {e} ¨ Die Uberlegung scheitert allerdings leider an dem inversen Element.

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Algebra und Zahlentheorie [Lecture notes] by Walter Gubler

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