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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" 
{TEXT -1 27 "Eine L\366sung zu Aufgabe (42)" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "(a) " }
{XPPEDIT 18 0 "f = x^5+3*x^2+6*x+3;" "6#/%\"fG,**$%\"xG\"\"&\"\"\"*&\"
\"$F)*$F'\"\"#F)F)*&\"\"'F)F'F)F)F+F)" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f:=x^5+
3*x^2+6*x+3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,**$)%\"xG\"\"&
\"\"\"F**&\"\"$F*)F(\"\"#F*F**&\"\"'F*F(F*F*F,F*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 13 "Maple sagt,  " }{XPPEDIT 18 0 "f;" "6#%\"fG" }
{TEXT -1 18 "  ist unzerlegbar." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "factor(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"&
\"\"\"F(*&\"\"$F()F&\"\"#F(F(*&\"\"'F(F&F(F(F*F(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 101 "Mit Hilfe des Eisensteinkriteriums best\344tigt s
ich dies direkt, wenn man das nicht sieht geht es auch " }}{PARA 0 "" 
0 "" {TEXT -1 8 "modular:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=2:" 
}}{PARA 0 "" 0 "" {TEXT -1 48 "Die unzerlegbaren Polynome vom Grad 2 m
od p sind" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "unz:=[]:for a0 in \{s
eq(k,k=0..p-1)\} do for a1 in \{seq(k,k=0..p-1)\}do \ng:=x^2+a1*x+a0:
\nif not 0 in \{seq(subs(x=t,g) mod p,t=0..p-1)\} then\nunz:=[op(unz),
g]; fi;od;od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unz;nops(unz);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7#,(*$)%\"xG\"\"#\"\"\"F)F'F)F)F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 87 "Die Divisionsreste von f mit unzerlegbaren Polynomen vom \+
Grad 2 sind von 0 verschieden:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 42 "seq(Rem(f,unz[k],x) mod p,k=1..nops(unz));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "f;" 
"6#%\"fG" }{TEXT -1 72 "  ist also unzerlegbar in Z[x] und somit auch \+
in Q[x] nach \2472 in Teil C." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "f = 5*x^4+4*x^3+3*x^
2+2*x+2;" "6#/%\"fG,,*&\"\"&\"\"\"*$%\"xG\"\"%F(F(*&F+F(*$F*\"\"$F(F(*
&F.F(*$F*\"\"#F(F(*&F1F(F*F(F(F1F(" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=5*x^
4+4*x^3+3*x^2+2*x+2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,,*&\"\"
&\"\"\")%\"xG\"\"%F(F(*&F+F()F*\"\"$F(F(*&F.F()F*\"\"#F(F(*&F1F(F*F(F(
F1F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Maple sagt,  " }{XPPEDIT 
18 0 "f;" "6#%\"fG" }{TEXT -1 18 "  ist unzerlegbar." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "factor(f);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,,*&\"\"&\"\"\")%\"xG\"\"%F&F&*&F)F&)F(\"\"$F&F&*&F,F&)F(\"\"#F&
F&*&F/F&F(F&F&F/F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "modular:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=3:" }}{PARA 0 "" 0 "" {TEXT -1 
48 "Die unzerlegbaren Polynome vom Grad 2 mod p sind" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 164 "unz:=[]:for a0 in \{seq(k,k=0..p-1)\} do for a1
 in \{seq(k,k=0..p-1)\}do \ng:=x^2+a1*x+a0:\nif not 0 in \{seq(subs(x=
t,g) mod p,t=0..p-1)\} then\nunz:=[op(unz),g]; fi;od;od;" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "unz;nops(unz);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7%,&*$)%\"xG\"\"#\"\"\"F)F)F),(F%F)F'F)F(F),(F%F)*&F(F)F'F)F)F(F
)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 69 "Die Divisionsreste von f mit unzerlegbaren Polynomen vom \+
Grad 2 sind:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq(Rem(f,u
nz[k],x) mod p,k=1..nops(unz));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,&%
\"xG\"\"\"F%F%,&F$F%\"\"#F%F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "
Das Ergebnis\"f  ist unzerlegbar\" von Maple ist demnach korrekt:" }}
{PARA 0 "" 0 "" {TEXT -1 117 "f hat keine Nullstellen mod 3, ist mod 3
 nicht durch ein quadratisches Polynom teilbar und ist somit unzerlegb
ar in  " }{XPPEDIT 18 0 "Z[3];" "6#&%\"ZG6#\"\"$" }{TEXT -1 5 "[x] ." 
}}{PARA 0 "" 0 "" {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 72 "  ist dan
n auch unzerlegbar in Z[x] und somit in Q[x] nach \2472 in Teil C." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 " (c)  " }
{XPPEDIT 18 0 "f = x^4+x^3+x^2+6*x+11;" "6#/%\"fG,,*$%\"xG\"\"%\"\"\"*
$F'\"\"$F)*$F'\"\"#F)*&\"\"'F)F'F)F)\"#6F)" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=x^
4+x^3+x^2+6*x+11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,,*$)%\"xG
\"\"%\"\"\"F**$)F(\"\"$F*F**$)F(\"\"#F*F**&\"\"'F*F(F*F*\"#6F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "modular: " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "erst bei p=31 wird " }{XPPEDIT 
18 0 "f;" "6#%\"fG" }{TEXT -1 22 " unzerlegbar mod p !!!" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 154 "p:=1:for k to 20 do p:=nextprime(p); FF:=(Fa
ctors(f) mod p)[2];print(FF);if nops(FF)=1 and FF[1][2]=1 then k:=20: \+
print('p'=p, `f ist unzerlegbar`);fi;od:" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7$7$,(*$)%\"xG\"\"$\"\"\"F*F(F*F*F*F*7$,&F(F*F*F*F*" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#7$7$,&%\"xG\"\"\"F'F'\"\"#7$,(*$)F&F(F'F'*&F(F'
F&F'F'F(F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$,&%\"xG\"\"\"\"\"%
F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$,(*$)%\"xG\"\"$\"\"\"F**&
\"\"&F*)F(\"\"#F*F*\"\"'F*F*7$,&F(F*F)F*F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%7$,&%\"xG\"\"\"\"\"#F'F'7$F&F'7$,&F&F'\"\"&F'F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$,&%\"xG\"\"\"\"\"%F'F'7$,(*$)F&\"
\"$F'F'*&\"#5F')F&\"\"#F'F'\"\"'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#7%7$,&%\"xG\"\"\"\"#:F'F'7$,(*$)F&\"\"#F'F'*&\"\")F'F&F'F'\"#8F'F'7$
,&F&F'\"#7F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$,**$)%\"xG\"\"$
\"\"\"F**&\"\"*F*)F(\"\"#F*F**&\"#;F*F(F*F*F*F*F*7$,&F(F*\"#6F*F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7$,&%\"xG\"\"\"\"\")F'F'7$,(*$)F&\"
\"#F'F'*&\"#6F'F&F'F'F-F'F'7$,&F&F'\"\"&F'F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$7$,**$)%\"xG\"\"$\"\"\"F**&\"\"(F*)F(\"\"#F*F**&\"#9F
*F(F*F*F)F*F*7$,&F(F*\"#BF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$,
,*$)%\"xG\"\"%\"\"\"F**$)F(\"\"$F*F**$)F(\"\"#F*F**&\"\"'F*F(F*F*\"#6F
*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"pG\"#J%2f~ist~unzerlegbarG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Nach Substitution x->x+1 wird
 allerdings das Eisensteinkriterium anwendbar:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g:=expand(subs(x=x+
1,f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,,*$)%\"xG\"\"%\"\"\"F
**&\"\"&F*)F(\"\"$F*F**&\"#5F*)F(\"\"#F*F**&\"#:F*F(F*F*\"#?F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "(d)  " }{XPPEDIT 18 0 "f = y^3*x^2
+6*y^3*x+9*y^3-y^2+y*x^2+5*y*x+6*y-x-7;" "6#/%\"fG,4*&%\"yG\"\"$%\"xG
\"\"#\"\"\"*(\"\"'F+*$F'F(F+F)F+F+*&\"\"*F+*$F'F(F+F+*$F'F*!\"\"*&F'F+
*$F)F*F+F+*(\"\"&F+F'F+F)F+F+*&F-F+F'F+F+F)F3\"\"(F3" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "restar:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
59 "f:=sort(expand((x+3)^2*y^3-y^2+(x+2)*(x+3)*y-(x+7)),[x,y]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,4*&)%\"xG\"\"#\"\"\")%\"yG\"\"$
F*F**(\"\"'F*F(F*F+F*F**&F'F*F,F*F**&\"\"*F*F+F*F**(\"\"&F*F(F*F,F*F**
$)F,F)F*!\"\"F(F7*&F/F*F,F*F*\"\"(F7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "collect(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&
,&%\"yG\"\"\"*$)F&\"\"$F'F'F')%\"xG\"\"#F'F'*&,(F'!\"\"*&\"\"&F'F&F'F'
*&\"\"'F'F)F'F'F'F,F'F'\"\"(F0*$)F&F-F'F0*&\"\"*F'F)F'F'*&F4F'F&F'F'" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "collect(f,y);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,,*&,(*$)%\"xG\"\"#\"\"\"F**&\"\"'F*F(F*F*\"
\"*F*F*)%\"yG\"\"$F*F**$)F/F)F*!\"\"*&,(F&F**&\"\"&F*F(F*F*F,F*F*F/F*F
*\"\"(F3F(F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "content(f,x
),content(f,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"F#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Substitution " }{XPPEDIT 18 0 "pro
c (x) options operator, arrow; 1 end proc;" "6#f*6#%\"xG7\"6$%)operato
rG%&arrowG6\"\"\"\"F*F*F*" }{TEXT -1 14 " und modulo 3:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=1,f) mod 3;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(*$)%\"yG\"\"$\"\"\"F(*&\"\"#F()F&F*F(F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "(e)   " }
{XPPEDIT 18 0 "f = x^4+y^2*x^3+(y+1)*x^2+1;" "6#/%\"fG,**$%\"xG\"\"%\"
\"\"*&%\"yG\"\"#F'\"\"$F)*&,&F+F)F)F)F)*$F'F,F)F)F)F)" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "f:=x^4+y^2*x^3+(y+1)*x^2+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"fG,**$)%\"xG\"\"%\"\"\"F**&)%\"yG\"\"#F*)F(\"\"$F*F**&,&F-F*F*F*F*)
F(F.F*F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Substitution " }
{XPPEDIT 18 0 "proc (y) options operator, arrow; 1 end proc;" "6#f*6#%
\"yG7\"6$%)operatorG%&arrowG6\"\"\"\"F*F*F*" }{TEXT -1 14 " und modulo
 2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(y=1,f) mod 2;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"gG,(*$)%\"xG\"\"%\"\"\"F**$)F(\"\"$F*F*F
*F*" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 
91 "  hat mod 2 keine Nullstellen und ist nicht durch  teilbar und som
it unzerlegbar. Also ist " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 41 "
   in Z[x] unzerlegbar und damit in Q[x]." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "oder besser noch: Substitu
tion " }{XPPEDIT 18 0 "proc (x) options operator, arrow; 1 end proc;" 
"6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"\"F*F*F*" }{TEXT -1 13 " un
d modulo 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(x=1,f)  \+
mod 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"yG\"\"#\"\"\"F(F&F(F
(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f;" "6#%
\"fG" }{TEXT -1 39 " ist somit mod 2 unzerlegbar. Also ist " }
{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 41 "   in Z[x] unzerlegbar und d
amit in Q[x]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 8 "(f) Ist " }{XPPEDIT 18 0 "f = y^4+y^3+y^2+y+1" "6#/%\"fG,,
*$%\"yG\"\"%\"\"\"*$F'\"\"$F)*$F'\"\"#F)F'F)F)F)" }{TEXT -1 17 " unzer
legbar in  " }{XPPEDIT 18 0 "F[4]*[y];" "6#*&&%\"FG6#\"\"%\"\"\"7#%\"y
GF(" }{TEXT -1 3 "  ?" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 " Dabei be
nutze ich " }{XPPEDIT 18 0 "F[4];" "6#&%\"FG6#\"\"%" }{TEXT -1 20 " in
 der Darstellung " }{XPPEDIT 18 0 "F[4] = \{0, 1, x, x+1\};" "6#/&%\"F
G6#\"\"%<&\"\"!\"\"\"%\"xG,&F+F*F*F*" }{TEXT -1 3 "  ." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
f:=y^4+y^3+y^2+y+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,,*$)%\"y
G\"\"%\"\"\"F**$)F(\"\"$F*F**$)F(\"\"#F*F*F(F*F*F*" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 17 "Zuerst mit Maple:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "alias(x=RootOf(x^2+x+1)):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "Factor(f,x) mod 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#*&,(*$)%\"yG\"\"#\"\"\"F)*&,&%\"xGF)F)F)F)F'F)F)F)F)F),(F%F)*&F,F)F'F
)F)F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f
;" "6#%\"fG" }{TEXT -1 28 " ist laut Maple unzerlegbar." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Eigene L
\366sung (ohne Factor):" }}{PARA 0 "" 0 "" {TEXT -1 20 "Die Elemente v
on F:=" }{XPPEDIT 18 0 "F[4];" "6#&%\"FG6#\"\"%" }{TEXT -1 27 " stelle
n wir wie folgt dar:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "F:=[0,1,x,x
+1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7&\"\"!\"\"\"%\"xG,&F(F'
F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Zuerst stellen wir fest, \+
ob " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 27 "  eine Nullstelle in \+
F hat:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for a in F do sim
plify(subs(y=a,f), RootOf) mod 2; od;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&%\"xG\"\"\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Demnach hat " }{XPPEDIT 
18 0 "f;" "6#%\"fG" }{TEXT -1 24 "  keine Nullstelle in F." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 64 "Nun bestimmen wir die unzerlegbaren Polyn
ome vom Grad 2 in F[y]:" }}{PARA 0 "" 0 "" {TEXT -1 41 "Die linearen P
olynome in y \374ber  F  sind:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "L
:=[seq(y+F[k],k=1..4)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7&%\"
yG,&F&\"\"\"F(F(,&F&F(%\"xGF(,(F&F(F*F(F(F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 53 "Die m\366glichen Produkte zweier linearer Polynome sind
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "L2:=\{\}:for k to 4 do for l \+
from k to 4 do \n     L2:=L2 union \{simplify(expand(L[k]*L[l]),RootOf
) mod 2\}; \nod;od;'L2'=L2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#L2G<
,*$)%\"yG\"\"#\"\"\",&F&F*F*F*,(F&F*F(F*F*F*,&F&F*F(F*,&F&F**&%\"xGF*F
(F*F*,(F&F*F/F*F(F*,*F&F*F/F*F(F*F0F*,(F&F*F*F*F0F*,&F&F*F0F*,*F&F*F/F
*F0F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Ihre Anzahl ist 10.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nops(L2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Die Menge \+
P2 aller Polynome in y vom Grad 2 \374ber F mit h\366chtem Koefiziente
n 1 ist:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "P2:=\{seq(seq(expand(y^
2+F[k]*y+F[l]),l=1..4),k=1..4)\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%#P2G<2*$)%\"yG\"\"#\"\"\",(F&F**&%\"xGF*F(F*F*F*F*,&F&F*F*F*,(F&F*F(F
*F*F*,&F&F*F(F*,&F&F*F,F*,(F&F*F,F*F(F*,*F&F*F,F*F(F*F-F*,(F&F*F*F*F-F
*,&F&F*F-F*,*F&F*F(F*F-F*F*F*,*F&F*F,F*F(F*F*F*,*F&F*F,F*F-F*F*F*,(F&F
*F(F*F-F*,(F&F*F,F*F-F*,,F&F*F,F*F(F*F-F*F*F*" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 24 "Es m\374ssen genau 16 sein:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "nops(P2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Die Menge der unzerlegbaren Polyno
me in F[y]  ist dann:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "U:
=map(collect,P2 minus L2,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG
<(,(*$)%\"yG\"\"#\"\"\"F+*&%\"xGF+F)F+F+F+F+,(F'F+*&,&F-F+F+F+F+F)F+F+
F+F+,*F'F+F)F+F-F+F+F+,*F'F+F/F+F-F+F+F+,(F'F+F)F+F-F+,(F'F+F,F+F-F+" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Es sind  6 an der Zahl." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(U);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Nun f
\374hren wir Probedivisionen durch, um festzustellen, ob eines der unz
erlegbaren Polynome aus U das gegebene Polynom  " }{XPPEDIT 18 0 "f;" 
"6#%\"fG" }{TEXT -1 8 "  teilt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 78 "l:=1:for k to 6 do Rem(f,U[k],y) mod 2; if %=0 then f||l:=U[k]
; l:=l+1;fi; od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Die Teiler vo
n f sind demnach:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seq(` \+
f`||j=f||j,j=1..l-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%$~f1G,(*$)%
\"yG\"\"#\"\"\"F**&%\"xGF*F(F*F*F*F*/%$~f2G,(F&F**&,&F,F*F*F*F*F(F*F*F
*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Probe" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "simplify(f-f||1*f||2, RootOf) mod 2;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "16" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }