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{SECT 0 {SECT 1 {PARA 256 "" 0 "" {TEXT -1 46 "Chapter 5\n\nPolynomial
s and Rational Functions\n" }}{PARA 0 "" 0 "" {TEXT 289 31 "\251 Copyr
ight 2003 by Andr\351 Heck." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 256 1 "1
" }{TEXT -1 35 ". Consider the rational expression " }{XPPEDIT 18 0 "(
x^4+x^3-4*x^2-4*x)/(x^4+x^3-x^2-x)" "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.
" }{TEXT -1 32 ". Transform the expression into\n" }{TEXT 265 3 "(a)" 
}{TEXT -1 2 "  " }{XPPEDIT 18 0 "(x+2)*(x+1)*(x-2)/(x^3+x^2-x-1)" "6#*
*,&%\"xG\"\"\"\"\"#F&F&,&F%F&F&F&F&,&F%F&F'!\"\"F&,**$F%\"\"$F&*$F%F'F
&F%F*F&F*F*" }{TEXT -1 2 "\n\n" }{TEXT 266 3 "(b)" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "(x^4+x^3-4*x^2-4*x)/(x*(x-1)*(x+1)^2" "6#*&,**$%\"xG\"
\"%\"\"\"*$F&\"\"$F(*&F'F(*$F&\"\"#F(!\"\"*&F'F(F&F(F.F(*(F&F(,&F&F(F(
F.F(,&F&F(F(F(F-F." }{TEXT -1 2 "\n\n" }{TEXT 267 3 "(c)" }{TEXT -1 1 
" " }{XPPEDIT 18 0 "(x+2)*(x-2)/(x-1)/(x+1)" "6#**,&%\"xG\"\"\"\"\"#F&
F&,&F%F&F'!\"\"F&,&F%F&F&F)F),&F%F&F&F&F)" }{TEXT -1 2 "\n\n" }{TEXT 
268 3 "(d)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2/(x-1)/(x+1) -4" "6#,&*
(%\"xG\"\"#,&F%\"\"\"F(!\"\"F),&F%F(F(F(F)F(\"\"%F)" }{XPPEDIT 18 0 "1
/(x-1)/(x+1)" "6#*(\"\"\"F$,&%\"xGF$F$!\"\"F',&F&F$F$F$F'" }{TEXT -1 
2 "\n\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 269 3 "(a)" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "expr := (x^4+x^3-4*x^2-4*x)/(x^4+x^3-x^2-x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG*&,**$)%\"xG\"\"%\"\"\"F+*$)F)
\"\"$F+F+*&F*F+)F)\"\"#F+!\"\"*&F*F+F)F+F2F+,*F'F+F,F+*$F0F+F2F)F2F2" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "factor(numer(expr))/denom
(expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"\"\"\"\"#!\"\"F&
,&F%F&F'F&F&,&F%F&F&F&F&,**$)F%\"\"$F&F&*$)F%F'F&F&F%F(F&F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "
" {TEXT 270 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "expr := (x^4+x^3-4*x^2
-4*x)/(x^4+x^3-x^2-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG*&,*
*$)%\"xG\"\"%\"\"\"F+*$)F)\"\"$F+F+*&F*F+)F)\"\"#F+!\"\"*&F*F+F)F+F2F+
,*F'F+F,F+*$F0F+F2F)F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 
"expand(numer(expr)) / factor(denom(expr));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#**,**$)%\"xG\"\"%\"\"\"F)*$)F'\"\"$F)F)*&F(F))F'\"\"#F)
!\"\"*&F(F)F'F)F0F)F'F0,&F'F)F)F0F0,&F'F)F)F)!\"#" }}}{PARA 0 "" 0 "" 
{TEXT -1 42 "A somewhat easier way to get the job done:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "applyop(factor, 2, expr);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#**,**$)%\"xG\"\"%\"\"\"F)*$)F'\"\"$F)F)*&F(F
))F'\"\"#F)!\"\"*&F(F)F'F)F0F)F'F0,&F'F)F)F0F0,&F'F)F)F)!\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "
" {TEXT 271 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "expr := (x^4+x^3-4*x^2
-4*x)/(x^4+x^3-x^2-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG*&,*
*$)%\"xG\"\"%\"\"\"F+*$)F)\"\"$F+F+*&F*F+)F)\"\"#F+!\"\"*&F*F+F)F+F2F+
,*F'F+F,F+*$F0F+F2F)F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"factor(expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"\"\"\"\"#
!\"\"F&,&F%F&F'F&F&,&F%F&F&F(F(,&F%F&F&F&F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 272 3 "(d)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 44 "expr := (x^4+x^3-4*x^2-4*x)/(x^4+x^3-x^2-x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG*&,**$)%\"xG\"\"%\"\"\"F+*
$)F)\"\"$F+F+*&F*F+)F)\"\"#F+!\"\"*&F*F+F)F+F2F+,*F'F+F,F+*$F0F+F2F)F2
F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(expr);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&%\"xG\"\"\"\"\"#!\"\"F&,&F%F&F'F&F
&,&F%F&F&F(F(,&F%F&F&F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,&%\"xG\"\"\"F'!
\"\"F(,&F&F'F'F'F(F&\"\"#F'*(\"\"%F'F%F(F)F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 168 "Remark: the
 ordering of terms may be different in your answers because Maple deci
des on its own grounds the term ordering and there is no way of having
 control on this." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 257 2 "2." }{TEXT 
-1 68 " Consider the same rational expression as in the previous exerc
ise.\n" }{TEXT 273 3 "(a)" }{TEXT -1 35 " Write it as a continued frac
tion.\n" }{TEXT 274 3 "(b)" }{TEXT -1 43 " Compute a partial fraction \+
decomposition.\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 275 3 "(a)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 43 "expr:= (x^4+x^3-4*x^2-4*x)/(x^4+x^3-x^2-x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG*&,**$)%\"xG\"\"%\"\"\"F+*
$)F)\"\"$F+F+*&F*F+)F)\"\"#F+!\"\"*&F*F+F)F+F2F+,*F'F+F,F+*$F0F+F2F)F2
F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "convert(expr, confrac
, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&\"\"$F$,&*$)%\"xG
\"\"#F$F$F$!\"\"F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}}{SECT 1 {PARA 5 "" 0 "" {TEXT 276 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "expr:= (x^4+x^3-4*x^2-4*x)/(x^4+x^3-x^2-x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%exprG*&,**$)%\"xG\"\"%\"\"\"F+*$)F)\"\"$F+F+*&F*F+)F
)\"\"#F+!\"\"*&F*F+F)F+F2F+,*F'F+F,F+*$F0F+F2F)F2F2" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "convert(expr, parfrac, x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*(\"\"$F$\"\"#!\"\",&%\"xGF$F$F$F(F$*(
F&F$F'F(,&F*F$F$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 258 2 "3." }{TEXT -1 5 " Let " }
{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 19 " be the polynomial " }
{XPPEDIT 18 0 "x^7+x^5+2*x^3+2*x^2+3*x+2" "6#,.*$%\"xG\"\"(\"\"\"*$F%
\"\"&F'*&\"\"#F'*$F%\"\"$F'F'*&F+F'*$F%F+F'F'*&F-F'F%F'F'F+F'" }{TEXT 
-1 2 ".\n" }{TEXT 277 3 "(a)" }{TEXT -1 8 " Factor " }{XPPEDIT 18 0 "f
" "6#%\"fG" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "Z[5]" "6#&%\"ZG6#\"\"
&" }{TEXT -1 11 ", and over " }{XPPEDIT 18 0 "Z[7]" "6#&%\"ZG6#\"\"(" 
}{TEXT -1 2 ".\n" }{TEXT 278 3 "(b)" }{TEXT -1 58 " How can you conclu
de from the former factorizations that " }{XPPEDIT 18 0 "f" "6#%\"fG" 
}{TEXT -1 109 " is irreducible over the ring of integers? (Hint: look \+
at the degrees of the factors.) Check with Maple that " }{XPPEDIT 18 
0 "f" "6#%\"fG" }{TEXT -1 28 " is indeed irreducible over " }{XPPEDIT 
18 0 "Z" "6#%\"ZG" }{TEXT -1 2 ".\n" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 
-1 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := x^7+x^5+2*x^3+2*x^2+3*x
+2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,.*$)%\"xG\"\"(\"\"\"F**$
)F(\"\"&F*F**&\"\"#F*)F(\"\"$F*F**&F/F*)F(F/F*F**&F1F*F(F*F*F/F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Factor(f) mod 5;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*&,(*$)%\"xG\"\"#\"\"\"F)*&\"\"$F)F'F)F)\"\"
%F)F),.*$)F'\"\"&F)F)*&F(F))F'F,F)F)*$)F'F+F)F)*&F,F)F&F)F)F'F)F+F)F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Factor(f) mod 7;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"\"\"%F&F&,0*$)F%\"\"'F&
F&*&\"\"$F&)F%\"\"&F&F&*&F-F&)F%F'F&F&*&\"\"#F&)F%F-F&F&*$)F%F3F&F&*&F
/F&F%F&F&F'F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 3 "(b)" }}{PARA 0 "" 0 "" {TEXT -1 
79 "The degrees of the factors in the previous part of the exercise ar
e 2, 5 (over " }{XPPEDIT 18 0 "Z[5]" "6#&%\"ZG6#\"\"&" }{TEXT -1 18 ")
, and 1, 6 (over " }{XPPEDIT 18 0 "Z[7]" "6#&%\"ZG6#\"\"(" }{TEXT -1 
247 "). Suppose that the polynomial is reducible over the ring of inte
gers. Then there exists a factor of degree 1, 2, or 3. A linear factor
 (degree = 1) would be present over any finite field, but as we see it
 does not occur in the factorization over " }{XPPEDIT 18 0 "Z[5]" "6#&
%\"ZG6#\"\"&" }{TEXT -1 62 ". A quadratic factor is not present in the
 factorization over " }{XPPEDIT 18 0 "Z[6]" "6#&%\"ZG6#\"\"'" }{TEXT 
-1 111 " and in this factorization there is only one linear factor. A \+
quadratic factor present in a factorization over " }{XPPEDIT 18 0 "Z" 
"6#%\"ZG" }{TEXT -1 139 " would be present in a factorization over any
 finite field or split into two linear factors. This does not occur in
 the factorization over " }{XPPEDIT 18 0 "Z[7]" "6#&%\"ZG6#\"\"(" }
{TEXT -1 345 ". So, the quadratic factor is not possible. What remains
 is a factor of degree 3. But then it would appear again in a factoriz
ation over any finite field or split into three linear factor or one q
uadratic and one linear factor. This does not occur in the factorizati
ons done before. Conclusion: the polynomial under consideration is irr
educible." }}{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(F-F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 260 
2 "4." }{TEXT -1 44 " Some factorizations modulo a prime number.\n" }
{TEXT 279 3 "(a)" }{TEXT -1 8 " Factor " }{XPPEDIT 18 0 "x^2-x" "6#,&*
$%\"xG\"\"#\"\"\"F%!\"\"" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "Z[2]" "
6#&%\"ZG6#\"\"#" }{TEXT -1 1 "\n" }{TEXT 280 3 "(b)" }{TEXT -1 8 " Fac
tor " }{XPPEDIT 18 0 "x^3-x" "6#,&*$%\"xG\"\"$\"\"\"F%!\"\"" }{TEXT 
-1 6 " over " }{XPPEDIT 18 0 "Z[3]" "6#&%\"ZG6#\"\"$" }{TEXT -1 1 "\n
" }{TEXT 281 3 "(c)" }{TEXT -1 8 " Factor " }{XPPEDIT 18 0 "x^5-x" "6#
,&*$%\"xG\"\"&\"\"\"F%!\"\"" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "Z[5]
" "6#&%\"ZG6#\"\"&" }{TEXT -1 1 "\n" }{TEXT 282 3 "(d)" }{TEXT -1 8 " \+
Factor " }{XPPEDIT 18 0 "x^23-x" "6#,&*$%\"xG\"#B\"\"\"F%!\"\"" }
{TEXT -1 6 " over " }{XPPEDIT 18 0 "Z[23]" "6#&%\"ZG6#\"#B" }{TEXT -1 
1 "\n" }{TEXT 283 3 "(e)" }{TEXT -1 33 " By now you may have an idea h
ow " }{XPPEDIT 18 0 "x^p-x" "6#,&)%\"xG%\"pG\"\"\"F%!\"\"" }{TEXT -1 
30 " factors for any prime number " }{XPPEDIT 18 0 "p" "6#%\"pG" }
{TEXT -1 108 "; check your conjecture for some prime number distinct f
rom previous choices. Do you understand the result?\n" }}{SECT 1 
{PARA 5 "" 0 "" {TEXT 284 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Factor(x
^2-x) mod 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"F&F&F&F
%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 
5 "" 0 "" {TEXT 285 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
"restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Factor(x^3-x) \+
mod 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"xG\"\"\"F&F&F&,&F%F&
\"\"#F&F&F%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 5 "" 0 "" {TEXT 286 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 
"Factor(x^5-x) mod 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,&%\"xG\"\"
\"\"\"$F&F&F%F&,&F%F&\"\"#F&F&,&F%F&F&F&F&,&F%F&\"\"%F&F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 
287 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Factor(x^23-x) mod 23;" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#*P%\"xG\"\"\",&F$F%\"\"&F%F%,&F$F%\"#9
F%F%,&F$F%\"#<F%F%,&F$F%\"#8F%F%,&F$F%\"#6F%F%,&F$F%\"#?F%F%,&F$F%F%F%
F%,&F$F%\"#7F%F%,&F$F%\"#=F%F%,&F$F%\"\"'F%F%,&F$F%\"\"(F%F%,&F$F%\"#>
F%F%,&F$F%\"#:F%F%,&F$F%\"#5F%F%,&F$F%\"#;F%F%,&F$F%\"\"#F%F%,&F$F%\"
\")F%F%,&F$F%\"\"*F%F%,&F$F%\"#@F%F%,&F$F%\"#AF%F%,&F$F%\"\"%F%F%,&F$F
%\"\"$F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 5 "" 0 "" {TEXT 288 3 "(e)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "for i fr
om 1 to 10 do\nFactor(x^(ithprime(i))-x) mod ithprime(i) end do;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\",&F$F%F%F%F%" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#*(%\"xG\"\"\",&F$F%\"\"#F%F%,&F$F%F%F%F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*,%\"xG\"\"\",&F$F%\"\"#F%F%,&F$F%\"\"
%F%F%,&F$F%F%F%F%,&F$F%\"\"$F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*0
%\"xG\"\"\",&F$F%\"\"#F%F%,&F$F%\"\"%F%F%,&F$F%F%F%F%,&F$F%\"\"'F%F%,&
F$F%\"\"$F%F%,&F$F%\"\"&F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*8,&%
\"xG\"\"\"\"\"(F&F&F%F&,&F%F&\"\"#F&F&,&F%F&\"#5F&F&,&F%F&\"\")F&F&,&F
%F&\"\"%F&F&,&F%F&F&F&F&,&F%F&\"\"'F&F&,&F%F&\"\"$F&F&,&F%F&\"\"*F&F&,
&F%F&\"\"&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*<,&%\"xG\"\"\"\"#7F
&F&,&F%F&\"\"(F&F&F%F&,&F%F&\"\"#F&F&,&F%F&\"#5F&F&,&F%F&\"\")F&F&,&F%
F&\"\"%F&F&,&F%F&F&F&F&,&F%F&\"#6F&F&,&F%F&\"\"'F&F&,&F%F&\"\"$F&F&,&F
%F&\"\"*F&F&,&F%F&\"\"&F&F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*D,&%\"
xG\"\"\"\"#7F&F&,&F%F&\"#;F&F&,&F%F&\"\"(F&F&F%F&,&F%F&\"\"#F&F&,&F%F&
\"#5F&F&,&F%F&\"\")F&F&,&F%F&\"\"%F&F&,&F%F&F&F&F&,&F%F&\"#8F&F&,&F%F&
\"#6F&F&,&F%F&\"\"'F&F&,&F%F&\"\"$F&F&,&F%F&\"#9F&F&,&F%F&\"\"*F&F&,&F
%F&\"#:F&F&,&F%F&\"\"&F&F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*H,&%\"x
G\"\"\"\"#7F&F&,&F%F&\"#;F&F&,&F%F&\"\"(F&F&F%F&,&F%F&\"\"#F&F&,&F%F&
\"#5F&F&,&F%F&\"\")F&F&,&F%F&\"\"%F&F&,&F%F&\"#=F&F&,&F%F&F&F&F&,&F%F&
\"#8F&F&,&F%F&\"#6F&F&,&F%F&\"#<F&F&,&F%F&\"\"'F&F&,&F%F&\"\"$F&F&,&F%
F&\"#9F&F&,&F%F&\"\"*F&F&,&F%F&\"#:F&F&,&F%F&\"\"&F&F&" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#*P,&%\"xG\"\"\"\"#7F&F&,&F%F&\"#;F&F&,&F%F&\"\"(F&
F&,&F%F&\"#>F&F&F%F&,&F%F&\"\"#F&F&,&F%F&\"#5F&F&,&F%F&\"\")F&F&,&F%F&
\"\"%F&F&,&F%F&\"#=F&F&,&F%F&F&F&F&,&F%F&\"#8F&F&,&F%F&\"#?F&F&,&F%F&
\"#6F&F&,&F%F&\"#<F&F&,&F%F&\"\"'F&F&,&F%F&\"\"$F&F&,&F%F&\"#9F&F&,&F%
F&\"\"*F&F&,&F%F&\"#:F&F&,&F%F&\"#AF&F&,&F%F&\"#@F&F&,&F%F&\"\"&F&F&" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#*fn,&%\"xG\"\"\"\"#CF&F&,&F%F&\"#7F&
F&,&F%F&\"#BF&F&,&F%F&\"#;F&F&,&F%F&\"\"(F&F&,&F%F&\"#>F&F&F%F&,&F%F&
\"#GF&F&,&F%F&\"\"#F&F&,&F%F&\"#DF&F&,&F%F&\"#5F&F&,&F%F&\"\")F&F&,&F%
F&\"\"%F&F&,&F%F&\"#=F&F&,&F%F&\"#EF&F&,&F%F&\"#FF&F&,&F%F&F&F&F&,&F%F
&\"#8F&F&,&F%F&\"#?F&F&,&F%F&\"#6F&F&,&F%F&\"#<F&F&,&F%F&\"\"'F&F&,&F%
F&\"\"$F&F&,&F%F&\"#9F&F&,&F%F&\"\"*F&F&,&F%F&\"#:F&F&,&F%F&\"#AF&F&,&
F%F&\"#@F&F&,&F%F&\"\"&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Ch
ecking for the first ten prime number we can assume the following conj
ecture" }}{PARA 0 "" 0 "" {TEXT 259 10 "Conjecture" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "x^p-x = product((x+k), k=0..p-1) mod p" "6#/,&)%\"xG%\"
pG\"\"\"F&!\"\"-%$modG6$-%(productG6$,&F&F(%\"kGF(/F1;\"\"!,&F'F(F(F)F
'" }{TEXT -1 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 8 "Suppose " }
{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 43 " is a root of the pol
ynomial. We show that " }{XPPEDIT 18 0 "alpha+1" "6#,&%&alphaG\"\"\"F%
F%" }{TEXT -1 17 " is also a root:\n" }{XPPEDIT 18 0 "(alpha+1)^p - (a
lpha+1)= alpha^p+p*alpha^(p-1)+ `...`+p*alpha+1-(alpha+1)" "6#/,&),&%&
alphaG\"\"\"F(F(%\"pGF(,&F'F(F(F(!\"\",.)F'F)F(*&F)F()F',&F)F(F(F+F(F(
%$...GF(*&F)F(F'F(F(F(F(,&F'F(F(F(F+" }{XPPEDIT 18 0 "``=alpha^p-alpha
 mod p" "6#/%!G-%$modG6$,&)%&alphaG%\"pG\"\"\"F*!\"\"F+" }{XPPEDIT 18 
0 "``=0" "6#/%!G\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "
Because 0 is a root, every integer less than " }{XPPEDIT 18 0 "p" "6#%
\"pG" }{TEXT -1 11 " is a root." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 261 2 "5." }{TEXT -1 5 " L
et " }{XPPEDIT 18 0 "f=2*x^4-3*x^2+x+4" "6#/%\"fG,**&\"\"#\"\"\"*$%\"x
G\"\"%F(F(*&\"\"$F(*$F*F'F(!\"\"F*F(F+F(" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "g=2*x^5-6*x^4-4*x^3+x^2-3*x-2" "6#/%\"gG,.*&\"\"#\"\"\"
*$%\"xG\"\"&F(F(*&\"\"'F(*$F*\"\"%F(!\"\"*&F/F(*$F*\"\"$F(F0*$F*F'F(*&
F3F(F*F(F0F'F0" }{TEXT -1 36 ". Consider them as polynomials over " }
{XPPEDIT 18 0 "Z[7]" "6#&%\"ZG6#\"\"(" }{TEXT -1 45 " and compute the \+
greatest common divisor of  " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 29 ". Determine also \+
polynomials " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "Z[7]
" "6#&%\"ZG6#\"\"(" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "s*f+t*g
=gcd(f,g)" "6#/,&*&%\"sG\"\"\"%\"fGF'F'*&%\"tGF'%\"gGF'F'-%$gcdG6$F(F+
" }{TEXT -1 22 " (gcd with respect to " }{XPPEDIT 18 0 "Z[7]" "6#&%\"Z
G6#\"\"(" }{TEXT -1 14 ", of course).\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 
"f := 2*x^4-3*x^2+x+4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,**&\"
\"#\"\"\")%\"xG\"\"%F(F(*&\"\"$F()F*F'F(!\"\"F*F(F+F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "g := 2*x^5-6*x^4-4*x^3+x^2-3*x-2;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,.*&\"\"#\"\"\")%\"xG\"\"&F(F(
*&\"\"'F()F*\"\"%F(!\"\"*&F/F()F*\"\"$F(F0*$)F*F'F(F(*&F3F(F*F(F0F'F0
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Gcd(f, g) mod 7;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"F(*&\"\"%F(F&F(F(
\"\"&F(" }}}{PARA 0 "" 0 "" {TEXT -1 60 "The extended gcd-computation \+
will determine the polynomials " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 11 " such that " }
{XPPEDIT 18 0 "s*f + t*g = gcd(f,g) mod 7" "6#/,&*&%\"sG\"\"\"%\"fGF'F
'*&%\"tGF'%\"gGF'F'-%$modG6$-%$gcdG6$F(F+\"\"(" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "Gcdex(f, g, x, 's', 't') mod 7;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"F(*&\"\"%F(F&F(F(\"\"&F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "s;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*&\"\"$\"\"\")%\"xG\"\"#F&F&*&\"\"'F&F(F&F&F+F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "t;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&\"\"%\"\"\"%\"xGF&F&\"\"'F&" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "Expand(s*f + t*g) mod 7;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"F(*&\"\"%F(F&F(F(\"\"&F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "
" {TEXT 263 2 "6." }{TEXT -1 28 " If you ask Maple to factor " }
{XPPEDIT 18 0 "x^2458+x^1229+1" "6#,(*$%\"xG\"%eC\"\"\"*$F%\"%H7F'F'F'
" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "Z" "6#%\"ZG" }{TEXT -1 255 ", t
hen the system will complain about insufficient memory, or after sever
al hours it will still not have found an answer. So, let us have a clo
ser look at this problem and see if we can assist Maple. First, we not
e that 1229 is a prime number (e.g., with " }{TEXT 0 7 "isprime" }
{TEXT -1 37 "). So, the polynomial is of the form " }{XPPEDIT 18 0 "x^
(2*p)+x^p+1" "6#,()%\"xG*&\"\"#\"\"\"%\"pGF(F()F%F)F(F(F(" }{TEXT -1 
8 ", where " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 102 " is a prime n
umber. Determine with Maple the factorization of this polynomial for l
ow prime values of " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 152 " and \+
propose a conjecture about the form of the factored polynomial for the
 general case.\nAn experienced mathematician can tell you that the pol
ynomial " }{XPPEDIT 18 0 "x^n-1" "6#,&)%\"xG%\"nG\"\"\"F'!\"\"" }
{TEXT -1 8 ", where " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 43 " is a
 natural number, can be factored over " }{XPPEDIT 18 0 "Z" "6#%\"ZG" }
{TEXT -1 31 " in the cyclotomic polynomials " }{XPPEDIT 18 0 "phi[k](x
)" "6#-&%$phiG6#%\"kG6#%\"xG" }{TEXT -1 23 ".\n                     " 
}{XPPEDIT 18 0 "x^n-1=product(phi[k](x), `k|n`)" "6#/,&)%\"xG%\"nG\"\"
\"F(!\"\"-%(productG6$-&%$phiG6#%\"kG6#F&%$k|grnG" }{TEXT -1 51 "\nThe
se cyclotomic polynomials are irreducible over " }{XPPEDIT 18 0 "Z" "6
#%\"ZG" }{TEXT -1 129 ". With this knowledge and under the assumption \+
that your conjecture is right, it should not be too difficult to prove
 the result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
80 "for i to 10 do\n  p := ithprime(i):\n  print('p'=p, factor(x^(2*p)
+x^p+1))\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"pG\"\"#*&,(*$
)%\"xGF%\"\"\"F+F*!\"\"F+F+F+,(F(F+F*F+F+F+F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$/%\"pG\"\"$,(*$)%\"xG\"\"'\"\"\"F+*$)F)F%F+F+F+F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"pG\"\"&*&,(*$)%\"xG\"\"#\"\"\"F,F*
F,F,F,F,,0*$)F*\"\")F,F,*$)F*\"\"(F,!\"\"*$)F*F%F,F,*$)F*\"\"%F,F4*$)F
*\"\"$F,F,F*F4F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"pG\"\"(*&,
(*$)%\"xG\"\"#\"\"\"F,F*F,F,F,F,,4*$)F*\"#7F,F,*$)F*\"#6F,!\"\"*$)F*\"
\"*F,F,*$)F*\"\")F,F4*$)F*\"\"'F,F,*$)F*\"\"%F,F4*$)F*\"\"$F,F,F*F4F,F
,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"pG\"#6*&,(*$)%\"xG\"\"#\"\"
\"F,F*F,F,F,F,,@*$)F*\"#?F,F,*$)F*\"#>F,!\"\"*$)F*\"#<F,F,*$)F*\"#;F,F
4*$)F*\"#9F,F,*$)F*\"#8F,F4*$)F*F%F,F,*$)F*\"#5F,F4*$)F*\"\"*F,F,*$)F*
\"\"(F,F4*$)F*\"\"'F,F,*$)F*\"\"%F,F4*$)F*\"\"$F,F,F*F4F,F,F," }}
{PARA 12 "" 1 "" {XPPMATH 20 "6$/%\"pG\"#8*&,(*$)%\"xG\"\"#\"\"\"F,F*F
,F,F,F,,D*$)F*\"#CF,F,*$)F*\"#BF,!\"\"*$)F*\"#@F,F,*$)F*\"#?F,F4*$)F*
\"#=F,F,*$)F*\"#<F,F4*$)F*\"#:F,F,*$)F*\"#9F,F4*$)F*\"#7F,F,*$)F*\"#5F
,F4*$)F*\"\"*F,F,*$)F*\"\"(F,F4*$)F*\"\"'F,F,*$)F*\"\"%F,F4*$)F*\"\"$F
,F,F*F4F,F,F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/%\"pG\"#<*&,(*$)%\"x
G\"\"#\"\"\"F,F*F,F,F,F,,P*$)F*\"#KF,F,*$)F*\"#JF,!\"\"*$)F*\"#HF,F,*$
)F*\"#GF,F4*$)F*\"#EF,F,*$)F*\"#DF,F4*$)F*\"#BF,F,*$)F*\"#AF,F4*$)F*\"
#?F,F,*$)F*\"#>F,F4*$)F*F%F,F,*$)F*\"#;F,F4*$)F*\"#:F,F,*$)F*\"#8F,F4*
$)F*\"#7F,F,*$)F*\"#5F,F4*$)F*\"\"*F,F,*$)F*\"\"(F,F4*$)F*\"\"'F,F,*$)
F*\"\"%F,F4*$)F*\"\"$F,F,F*F4F,F,F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6
$/%\"pG\"#>*&,(*$)%\"xG\"\"#\"\"\"F,F*F,F,F,F,,TF,F,F*!\"\"*$)F*\"\"%F
,F.*$)F*\"\"$F,F,*$)F*\"#5F,F.*$)F*\"\"*F,F,*$)F*\"\"'F,F,*$)F*\"\"(F,
F.*$)F*\"#8F,F.*$)F*\"#7F,F,*$)F*\"#?F,F.*$)F*\"#;F,F.*$)F*\"#CF,F,*$)
F*\"#@F,F,*$)F*\"#IF,F,*$)F*\"#FF,F,*$)F*\"#LF,F,*$)F*\"#OF,F,*$)F*\"#
NF,F.*$)F*\"#EF,F.*$)F*\"#BF,F.*$)F*\"#=F,F,*$)F*\"#:F,F,*$)F*\"#HF,F.
*$)F*\"#KF,F.F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/%\"pG\"#B*&,(*$)%
\"xG\"\"#\"\"\"F,F*F,F,F,F,,jnF,F,*$)F*\"#WF,F,*$)F*\"#SF,!\"\"*$)F*\"
#VF,F4*$)F*\"#TF,F,F*F4*$)F*\"\"%F,F4*$)F*\"\"$F,F,*$)F*\"#5F,F4*$)F*
\"\"*F,F,*$)F*\"\"'F,F,*$)F*\"\"(F,F4*$)F*\"#8F,F4*$)F*\"#AF,F4*$)F*\"
#7F,F,*$)F*\"#MF,F4*$)F*\"#;F,F4*$)F*\"#>F,F4*$)F*\"#@F,F,*$)F*\"#NF,F
,*$)F*\"#EF,F,*$)F*\"#DF,F4*$)F*F%F,F,*$)F*\"#=F,F,*$)F*\"#:F,F,*$)F*
\"#QF,F,*$)F*\"#PF,F4*$)F*\"#HF,F,*$)F*\"#KF,F,*$)F*\"#GF,F4*$)F*\"#JF
,F4F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/%\"pG\"#H*&,(*$)%\"xG\"\"#\"
\"\"F,F*F,F,F,F,,joF,F,*$)F*\"#WF,F,*$)F*\"#SF,!\"\"*$)F*\"#VF,F4*$)F*
\"#TF,F,*$)F*\"#`F,F,*$)F*\"#]F,F,*$)F*\"#ZF,F,*$)F*\"#cF,F,*$)F*\"#_F
,F4*$)F*\"#bF,F4*$)F*\"#\\F,F4F*F4*$)F*\"\"%F,F4*$)F*\"\"$F,F,*$)F*\"#
5F,F4*$)F*\"\"*F,F,*$)F*\"\"'F,F,*$)F*\"\"(F,F4*$)F*\"#8F,F4*$)F*\"#AF
,F4*$)F*\"#7F,F,*$)F*\"#MF,F4*$)F*\"#;F,F4*$)F*\"#>F,F4*$)F*\"#CF,F,*$
)F*\"#@F,F,*$)F*\"#FF,F,*$)F*\"#NF,F,*$)F*\"#DF,F4*$)F*\"#=F,F,*$)F*\"
#:F,F,*$)F*\"#QF,F,*$)F*\"#PF,F4*$)F*F%F,F,*$)F*\"#KF,F,*$)F*\"#GF,F4*
$)F*\"#JF,F4*$)F*\"#YF,F4F," }}}{PARA 0 "" 0 "" {TEXT 262 10 "Conjectu
re" }{TEXT -1 18 " For prime number " }{XPPEDIT 18 0 "p*`>`*3" "6#*(%
\"pG\"\"\"%\">GF%\"\"$F%" }{TEXT -1 28 ", one has the factorization " 
}{XPPEDIT 18 0 "x^(2*p)+x^p+1=(x^2+x+1)*phi[3*p](x)" "6#/,()%\"xG*&\"
\"#\"\"\"%\"pGF)F))F&F*F)F)F)*&,(*$F&F(F)F&F)F)F)F)-&%$phiG6#*&\"\"$F)
F*F)6#F&F)" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "phi[k](x) " "6#-&%$
phiG6#%\"kG6#%\"xG" }{TEXT -1 36 " is the k-th  cyclotomic polynomial.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "First
 we verify the conjecture with some Maple calculations:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numtheory):" }}{PARA 7 "" 1 "
" {TEXT -1 69 "Warning, the protected name order has been redefined an
d unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "\{ seq(
\n testeq( factor(x^(2*ithprime(i))+x^ithprime(i)+1) =\n         (x^2+
x+1)*cyclotomic(3*ithprime(i),x) ),\n i = 3..25 ) \};" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 28 "The pro
of of the conjecture:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "x^(3*p)-1 = (x^(2*p)+x^p+1)*(x^p-1)" "6#/,&)%\"xG*&\"\"$\"\"\"%\"pG
F)F)F)!\"\"*&,()F&*&\"\"#F)F*F)F))F&F*F)F)F)F),&)F&F*F)F)F+F)" }{TEXT 
-1 36 ". Hence, the prime factorization of " }{XPPEDIT 18 0 "x^(2*p)+x
^p+1" "6#,()%\"xG*&\"\"#\"\"\"%\"pGF(F()F%F)F(F(F(" }{TEXT -1 29 " equ
als the factorization of " }{XPPEDIT 18 0 "x^(3*p)-1" "6#,&)%\"xG*&\"
\"$\"\"\"%\"pGF(F(F(!\"\"" }{TEXT -1 27 " divided by the factors of " 
}{XPPEDIT 18 0 "x^p-1)" "6#,&)%\"xG%\"pG\"\"\"F'!\"\"" }{TEXT -1 34 ".
 Therefore, the factorization of " }{XPPEDIT 18 0 "x^(2*p)+x^p+1" "6#,
()%\"xG*&\"\"#\"\"\"%\"pGF(F()F%F)F(F(F(" }{TEXT -1 42 " is the produc
t of cyclotomic polynomials " }{XPPEDIT 18 0 "phi[k](x)" "6#-&%$phiG6#
%\"kG6#%\"xG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "k" "6#%\"kG" }
{TEXT -1 9 " divides " }{XPPEDIT 18 0 "3*p" "6#*&\"\"$\"\"\"%\"pGF%" }
{TEXT -1 10 " and not  " }{XPPEDIT 18 0 "p" "6#%\"pG" }{TEXT -1 6 ". S
o, " }{XPPEDIT 18 0 "k=3" "6#/%\"kG\"\"$" }{TEXT -1 11 " (assuming " }
{XPPEDIT 18 0 "p<>3" "6#0%\"pG\"\"$" }{TEXT -1 5 ") or " }{XPPEDIT 18 
0 "k=3*p" "6#/%\"kG*&\"\"$\"\"\"%\"pGF'" }{TEXT -1 11 ". Because  " }
{XPPEDIT 18 0 "phi[3](x)=x^2+x+1" "6#/-&%$phiG6#\"\"$6#%\"xG,(*$F*\"\"
#\"\"\"F*F.F.F." }{TEXT -1 39 " we have the formula in the conjecture.
" }}{PARA 0 "" 0 "" {TEXT -1 47 "Just to the correctness in the partic
ular case:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "p := 1229;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"%H7" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 36 "expand((x^2+x+1)*cyclotomic(3*p,x));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*$)%\"xG\"%H7F$F$*$)F'\"%eCF$F$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sort(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*$)%\"xG\"%eC\"\"\"F(*$)F&\"%H7F(F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0" 0 }{VIEWOPTS 1 1 0 
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