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Algebra fur Informatiker by Buchmann J. PDF

By Buchmann J.

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Satz Ist G endlich, so folgt ajGj = 1 fur alle a 2 G. Beweis: ordG a = jhaij = minfx 2 IN : ax = 1g teilt jGj Satz=)2:15:7 ajGj = (aordG a)(G:hai) = 1. 9. Ubung Satz von Fermat, Fermattest, Pseudoprimzahl. 10. Satz Jede Gruppe von Primzahlordnung ist zyklisch. 11. Satz Ist G zyklisch und ist a ein Erzeuger von G, so ist hadi fur alle Teiler d von jGj die eindeutig bestimmte Untergruppe von G mit Index d. Ist G unendlich, so ist hei die eindeutig bestimmte Untergruppe von G mit Index 1. 16 Anwendung des Satzes von Lagrange Sei G eine endliche Gruppe.

Newton Interpolation: Sei (ai)i 0 eine Folge von Ringelementen. Wir konstruieren induktiv Polynome pk (x) = c0 + c1(x ; a0 ) + c2 (x ; a0)(x ; a1) + : : : + ck (x ; a0 ) mit pk (ai) = bi (x ; ak;1 ) 0 i k: Hierzu geht man folgenderma en vor: b0 b1 b2 b3 b4 .. b01 b12 b23 b34 b02 b13 b24 b03 b14 b04 bij = bi ajj;;baii j; j > i +1 1 und b0 = c0 b01 = c1 b02 = c2 b03 = c3 ... Vorteil dieser Methode: Es ist leicht zusatzliche Stutzstellen dazuzunehmen. Version 6. 1. De nition Sei R Integritatsbereich.

Sei g = ggT(f f 0) 62 K: ) gjf gjf 0 ) f = gh f 0 = g0h + gh0 ) gjg0h. Da "deg( g0) < deg(g ) folgt: g := ggT(g h) 2= K , ) gjg gjh ) g 2jf . 3. Beispiel K = IFp = ZZ=pZZ, p = 2, f (x) = x4 + x3 + x + 1, f 0(x) = x2 + 1. Berechne ggT(f (x) f 0(x)): (x4 + x3 + x + 1) : (x2 + 1) = x2 + x + 1 ) ggT(f (x) f 0(x)) = f 0(x) 2= K ) f (x) nicht quadratfrei, 2 2 f (x) = (x + 1)(x + x + 1) = (x + 1)2 (x2 + x + 1) Als Erstes werden wir untersuchen, wie man eine quadratfreie Zerlegung eines Polynoms berechnen kann.

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