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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 57 "Beispiele von  Berechnung
en von Primzerlegungen mit Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 72 "Dass die dabei auftretenden Ringe faktori
ell sind ergibt sich mit Hilfe " }}{PARA 0 "" 0 "" {TEXT -1 25 "der Th
eorie aus Teil C \2471" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 17 "Z ist euklidisch:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ifactor(10^20-10);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 20 "Q[x] ist euklidisch:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor(x^20-1024);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
35 "(Z/pZ)[x] , p prim, ist euklidisch:" }}{PARA 0 "" 0 "" {TEXT -1 4 
"p=7:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "f:=x^7+2*x^6+3*x^5+4*x^4+5*x^3+6*x^2+1;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 50 "Factor(x^7+2*x^6+3*x^5+4*x^4+5*x^3+6*x^2+1) mod \+
7;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Q[x,y,z] ist faktoriell:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f:
=-x^2*y-z^2*x+z*x^2-y^2*z+x*y^2+y*z^2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "factor(f);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Z[
i] ist euklidisch (das war in der Vorlesung nicht dran):" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(GaussIn
t):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "GIfactor(2*I);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "Z[i][x] ist faktoriell:" }}{PARA 0 "" 0 "
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" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "GIfacpoly(x^20+1);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 257 25 "Zusatz f\374r Interessierte:" }{TEXT -1 
3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 
"Aus Maple Help f\374r factorEQ:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 76 "\"Euclidean quadratic number fields have \+
been completely determined. They are" }}{PARA 0 "" 0 "" {TEXT -1 78 " \+
 Z(sqrt(d)) where d = -1, -2, -3, -7, -11, 2, 3, 5, 6, 7, 11, 13, 17, \+
19, 21," }}{PARA 0 "" 0 "" {TEXT -1 29 "  29, 33, 37, 41, 57, and 73.
" }}{PARA 0 "" 0 "" {TEXT -1 83 "             (Nach Scheja/Storch Teil
2,S.159 Mitte trifft dies nur zu, wenn man die" }}{PARA 0 "" 0 "" 
{TEXT -1 59 "              Norm   als euklidische Funktion vorschreibt
.)" }}{PARA 0 "" 0 "" {TEXT -1 78 "- When d = 2, 3 (mod 4), all intege
rs of Z(sqrt(d)) have the form a+b*sqrt(d)," }}{PARA 0 "" 0 "" {TEXT 
-1 71 "  where a and b are rational integers.  When d = 1 (mod 4), int
egers of" }}{PARA 0 "" 0 "" {TEXT -1 73 "  Z(sqrt(d)) are of the form \+
1/2*(a+b*sqrt(d)) where a and b are rational" }}{PARA 0 "" 0 "" {TEXT 
-1 35 "  integers and of the same parity.\"" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Es folgen einige Beispiele hier
zu:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "with(numtheory):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "factorEQ(
10^20-10,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factorEQ(10^20-10,73);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Z[Wurzel aus 2][x] ist faktoriell:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "alias(alpha=RootOf(x^2-2)):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evala(Factor(x^20-1024,alpha));" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "(Z/7Z)[Wurzel von x^7+x^3+1] ist \+
einK\366rper," }}{PARA 0 "" 0 "" {TEXT -1 47 "(Z/7Z)[Wurzel von x^7+x^
3+1][x] ist euklidisch:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "alias(be
ta=RootOf(x^7+x^3+1));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Factor(x^
7+x^3+1,beta) mod 7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "`mo
d`:=mods:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Factor(x^7+x^3+1 ,beta
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