Algebra fur Informatiker by Buchmann J. PDF

By Buchmann J.

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GgT(a p) = 1. Dann kann man 1 = xa + yp schreiben, also b = xab + ypb, woraus folgt, da p ein Teiler von b ist. 8. De nition R hei t ZPE-Ring, wenn jede von Null verschiedene Nichteinheit von R ein Produkt irreduzibler Elemente ist und die in diesem Produkt vorkommenden irreduziblen Elemente bis auf Assoziiertheit eindeutig bestimmt sind. Version 6. 9. Beispiel ZZ ist ZPE-Ring. 10. Satz Folgende Aussagen sind aquivalent. 1. R ist ZPE-Ring. 2. Jede von Null verschiedene Nichteinheit ist Produkt irreduzibler Elemente und jedes irreduzible Element ist Primelement.

Dann gibt es genau einen nicht trivialen hf i-Orbit Y , auf dem g nicht die Identitat ist. Hierfur gilt g = fY . 9. Beispiel f = 13 22 34 48 56 65 77 81 hf i-Orbits: f1| 3{z4 8}g f2g f|{z} 5 6 g f7g Y1 ! 10. Satz Die Anzahl der Permutationen, in die eine Permutation faktorisiert werden kann, ist stets gerade oder stets ungerade. Beweis: Fur f 2 Sn setze Y f (i) ; f (j ) : i ; j 1 i

Satz In einem Hauptidealring sind die von f0g verschiedenen Primideale maximal. Beweis: Sei pR Primideal, pR echt enthalten in bR. Dann ist b echter Teiler von p (vgl. 2), also b = 1. Version 6. 12 Faktorisierung von Polynomen In diesem Abschnitt wollen wir uns damit beschaftigen, wie man Polynome faktorisieren kann. Sei dazu K Korper dann ist K x] euklidisch. ) K x] ist Hauptidealring ) K x] ist ZPE-Ring. Sei f 2 K x]. 1. De nition f 2 K x] hei t quadratfrei, wenn in obiger Zerlegung von f alle ei = 1 sind und fi 6= fj 8i 6= j .

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Algebra fur Informatiker by Buchmann J.


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