Algebra I.K. by Hans Kurzweil PDF

By Hans Kurzweil

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Extra resources for Algebra I.K.

Example text

Zweitens berechnet er Polynome P, Q , so dass P · N = Q · A und ggT(P, Q) = 1 . 1 (Seite 37) ist dann P · N = kgV(A, N ) . Die Bestimmung dieser Polynome P, Q ist fu¨r das Funktionieren eines ReedSolomon Codes wichtig. Dies erkl¨aren wir im letzten Kapitel. 8. Wir setzen daher im Folgenden grad A < n , n = grad N voraus. Der EA bestimmt G : = ggT(N, A) . Wir ku¨mmern uns wie schon in Kap. 3 nicht um die Normierung von G, jedes skalare Vielfache = 0 von G bezeichnen wir ebenfalls mit G; analog verfahren wir mit dem kgV(A, N ).

Also ist FN = {a0 + a1 x | a0 , a1 ∈ R} und x2 = −1 . Die Multiplikation ist (a0 + a1 x)(b0 + b1 x) = (a0 b0 − a1 b1 ) + (a0 b1 + a1 b0 )x . Setzt man i = x, so ist i2 = −1 und FN = {a0 +i a1 | a0 , a1 ∈ R}. Die Addition und Multiplikation zeigen, dass FN der K¨orper C der komplexen Zahlen ist. In den n¨achsten vier Beispielen ist F = Z2 = {0, 1}; in F ist 1 = −1, die Addition f¨allt also mit der Subtraktion zusammen. Schreibt man a0 a1 fu¨r a0 + a1 x, so ist FN = {00, 10, 01, 11} , dabei ist 00 das Null– und 01 das Einselement.

Ist A nicht irreduzibel, so existieren nichttriviale Teiler B, C von A mit A = B · C, also ist grad B, grad C < grad A. Per Induktion nach grad A k¨onnen wir P bP , C = B= P ∈P(B) C cP P ∈P(C) 41 annehmen. Beachtet man PRIM, also P(A) = P(B) ∪ P(C) , so folgt (∗). Wir behaupten nun weiter, dass die Exponenten aP in (∗) eindeutig durch A festgelegt sind: Sei auch A= P ∈P(A) P aP , a P ∈ N , und etwa R ∈ P(A) mit aR = aR . Wir schreiben    A = R aR  P ∈P(A)\{R} P aP  = R aR  P ∈P(A)\{R}  P aP  .

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Algebra I.K. by Hans Kurzweil


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