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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 257 17 "Beispiel 17.8 (b)" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "E
in Polynom, das durch iteriertes Wurzelziehen aufl\366sbar ist:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}{MPLTEXT 1 0 17 "f:=x^12-10*x^6+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"fG,(*$)%\"xG\"#7\"\"\"F**&\"#5F*)F(\"\"'F*!\"\"F*F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "Ist f unzerlegbar ?" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 10 "factor(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,(*$)%\"xG\"#7\"\"\"F(*&\"#5F()F&\"\"'F(!\"\"F(F(" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 25 "Maple sagt demnach \"ja\". " }}{PARA 0 "" 0 "" 
{TEXT -1 125 "Dass Maple dies so schnell entscheiden kann beruht auf T
heorie und Verfahren des entsprechenden Teilgebietes der sogenannten \+
" }{TEXT 258 15 "Computeralgebra" }{TEXT -1 36 ".Ein Standardwerk dazu
 ist das Buch " }{TEXT 259 71 "\"Modern Computer Algebra\" von Joachim
 von zur Gathen und J\374rgen Gerhard" }{TEXT -1 34 ".Die Internetseit
e zum Buch ist:  " }{URLLINK 17 "Modern Computer Algebra" 4 "http://ww
w-math.uni-paderborn.de/mca/" "" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor(f,sq
rt(2));" }}{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)F)F/F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "factor(f,sqrt(3));" }}{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.!\"\"F)F)F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "factor(f,\{sqrt(2),sqrt(3)\});" }}{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.F)F),(F%F)F*F)F.F)F)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 37 "alias(alpha=(sqrt(2)+sqrt(3))^(1/3));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%&alphaG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "ff:=simplify(factor(f,\{sqrt(2),sqrt(3),alpha\}));" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#ffG*0,(*$)%\"xG\"\"#\"\"\"F+*&F)F+
%&alphaGF+F+*$),&*$\"\"$#F+F*F+*$F*F3F+#F*F2F+F+F+,,F'F+*(F/F+F)F+F2F3
!\"\"*(F/F+F)F+F*F3F+*&F-F+F2F3F+*&F-F+F*F3F8F+,,F'F+F7F+F9F8F:F+F;F8F
+,(F'F+F,F8F.F+F+,(F)F+*&F/F+F2F3F+*&F/F+F*F3F8F+,(F)F+F?F8F@F+F+,&F'F
+F.F8F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "g:=op(1,ff);h:=o
p(2,ff);k:=op(3,ff); l:=op(4,ff);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"gG,(*$)%\"xG\"\"#\"\"\"F**&F(F*%&alphaGF*F**$),&*$\"\"$#F*F)F**$F)F
2F*#F)F1F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,,*$)%\"xG\"\"#
\"\"\"F**(),&*$\"\"$#F*F)F**$F)F0F*#F)F/F*F(F*F/F0!\"\"*(F,F*F(F*F)F0F
**&%&alphaGF*F/F0F**&F6F*F)F0F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"kG,,*$)%\"xG\"\"#\"\"\"F**(),&*$\"\"$#F*F)F**$F)F0F*#F)F/F*F(F*F/F0F
**(F,F*F(F*F)F0!\"\"*&%&alphaGF*F/F0F**&F6F*F)F0F4" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"lG,(*$)%\"xG\"\"#\"\"\"F**&F(F*%&alphaGF*!\"\"*$)
,&*$\"\"$#F*F)F**$F)F3F*#F)F2F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "collect(h,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$
)%\"xG\"\"#\"\"\"F(*&,&*&),&*$\"\"$#F(F'F(*$F'F0F(#F'F/F(F/F0!\"\"*&F,
F(F'F0F(F(F&F(F(*&%&alphaGF(F'F0F3*&F6F(F/F0F(" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Maple erkennt nicht \+
 " }{XPPEDIT 18 0 "alpha^2;" "6#*$%&alphaG\"\"#" }{TEXT -1 92 " in die
sen Ausdr\374cken !  Die Nullstellen des letzten quadratischen Polynom
s sind laut Maple:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 27 "so:=[solve(%)]:so[1];so[2];" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(*(\"\"#!\"\",&*$\"\"$#\"\"\"F%F+*$F%F*F+#F%F)F)F
*F+*(F%F&F'F-F%F*F&*&^#F*F+,**&\"\"&F+)F'#\"\"%F)F+F&**F%F+F4F+F)F*F%F
*F+*(F6F+%&alphaGF+F%F*F&*(F6F+F9F+F)F*F+F*F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*(\"\"#!\"\",&*$\"\"$#\"\"\"F%F+*$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+F5F+F)F*F%F*F
+*(F7F+%&alphaGF+F%F*F&*(F7F+F:F+F)F*F+F*F+" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 233 "In diesen Ausdr\374cke
n taucht die komplexe Zahl i (bei Maple I) auf. Dass komplexe L\366sun
gen vorliegen h\344tte man besser vorher erkundet. Mit dieser Informat
ion ergibt sich n\344mlich wie folgt eine vollst\344ndige Zerlegung in
 Linearfaktoren:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "ff:=simplify(factor(f,\{sqrt(2),sqrt(3),alpha,I\}));
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#ffG,$*:\"$c#!\"\",(*&\"\"#\"\"
\"%\"xGF,F,%&alphaGF,*(^#F(F,F.F,\"\"$#F,F+F,F,,(*&F+F,F-F,F,F.F,*(F.F
,F1F2^#F,F,F,F,,,*&F+F,F-F,F,*&),&*$F1F2F,*$F+F2F,#F+F1F,F1F2F(*&F:F,F
+F2F,**F1F2F+F2F:F,F6F,F,*&^#!\"$F,F:F,F,F,,,*&F+F,F-F,F,F9F(F?F,*&^#F
1F,F:F,F,**F0F,F1F2F+F2F:F,F,F,,,*&F+F,F-F,F,F9F,F?F(F@F,FAF,F,,,*&F+F
,F-F,F,F9F,F?F(FFF,FHF,F,,(*&F+F,F-F,F,F.F(F5F,F,,(*&F+F,F-F,F,F.F(F/F
,F,,(F-F,F9F,F?F(F,,(F-F,F9F(F?F,F,,&*$)F-F+F,F,*$F:F,F(F,F," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Te
ilprobe durch Einsetzen der zuerst aufgef\374hrten Nullstelle:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "sim
plify(subs(x=-(alpha+I*alpha*sqrt(3))/2,f));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "An den v
orangegangen Rechnung erkennt man dass ein Zerf\344llungsk\366rper von
 f \374ber Q den Grad  24 hat. Dies kann man auch wie folgt best\344ti
gen:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "alias(beta=RootOf(f));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$%&alphaG%%betaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fa
ctor(f,beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*2,,*$)%\"xG\"\"#\"\"
\"F)*(\"#5F)F'F))%%betaG\"\"&F)!\"\"*&F'F))F-\"#6F)F)*&F+F))F-\"\"%F)F
)*$)F-F+F)F/F),,F%F)*(F+F)F'F)F,F)F)F0F/*&F+F)F4F)F)F6F/F),(F%F)*&F'F)
F-F)F/*$)F-F(F)F)F),(F%F)F<F)F=F)F),(F'F)*&F+F)F,F)F/*$F1F)F)F),(F'F)*
&F+F)F,F)F)FBF/F),&F'F)F-F)F),&F'F)F-F/F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 69 "factor(f,\{beta,R
ootOf(x^2+10*x*beta^5-x*beta^11+10*beta^4-beta^10)\});" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#*:,&%\"xG\"\"\"-%'RootOfG6#,**$)%#_ZG\"\"#F&F&*&,&
*&\"#5F&)%%betaG\"\"&F&F&*$)F4\"#6F&!\"\"F&F-F&F&*$)F4F2F&F9*&F2F&)F4
\"\"%F&F&F&F&,**&F2F&F3F&F9F6F&F'F9F%F&F&,**&F2F&F3F&F&F6F9F'F&F%F&F&,
&F%F&F'F9F&,(F%F&*&F2F&F3F&F9F6F&F&,(F%F&*&F2F&F3F&F&F6F9F&,(F4F9*&F'F
&)F4F.F&F9F%F&F&,&FIF&F%F&F&,&FIF9F%F&F&,(F4F&FIF&F%F&F&,&F%F&F4F&F&,&
F%F&F4F9F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "16 1 0" 61 }{VIEWOPTS 0 0 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }