Aufbaukurs Mathematik by Kunz Ernst

By Kunz Ernst

Kunz E. Algebra (3ed, Vieweg, 2000)(de)(ISBN 3528272430)-o

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D. Korollar. a) Z X ; : : : ; Xn] ist ein faktorieller Ring. b) Fur jeden Korper K ist K X ; : : : ; Xn] faktoriell. V. Quotientenringe Die Bildung von Quotientenringen ist eine Verallgemeinerung der Konstruktion der rationalen aus den ganzen Zahlen. Es entstehen zahlreiche neue Ringe, unter denen auch viele faktoriell sind. Eine Teilmenge N eines Rings R hei t multiplikativ abgeschlossen, wenn gilt: Es ist 1 2 N und fur alle a; b 2 N ist auch a É b 2 N . Beispiele multiplikativ abgeschlossener Teilmengen von R sind: a) Die Menge der Einheiten von R .

In Z ersetzt man Grad-Argumente durch entsprechende Argumente fur den Absolutbetrag. Fur f; g 2 K X ] n f0g hat man eine Kette von Gleichungen, die sich jeweils durch Division mit Rest ergeben: f =q Ég+r g =q Ér +r r = q Ér +r .. rn = qn É rn + rn rn = qn É rn 1 2 (3) 1 3 1 1 2 2 2 1 3 1 deg r < deg g deg r < deg r deg r < deg r .. deg rn < deg rn 1 2 1 3 2 1 +1 Da der Grad des Divisionsrestes bei jedem Schritt abnimmt, mu die Division nach endlich vielen Schritten schlie lich aufgehen. Satz.

Unter den Voraussetzungen von Aufgabe 14) sei SpL=K (a) die Spur des Endomorphismus a und NL=K (a) := det a . Die Abbildungen SpL=K : L ! K ; NL=K : L ! K hei en Spur bzw. Norm der endlichen Korpererweiterung L=K . h. NL=K (ab) = NL=K (a) É NL=K (b). c) Ist a = X t + t 1X t 1 + É É É + 0 , so ist SpL=K (a) = t 1 ; NL=K (a) = ( 1)t 0 d) Ist fa = X n + n 1X n 1 + É É É + 0 (i 2 K ) das Minimalpolynom von a uber K und L : K (a)] =: m , L : K ] =: t , so gilt p SpL=K (a) = m É n 1 ; NL=K (a) = ( 1)t 0m p 16) Sei K = Q ( d) mit d 2 Q .

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