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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:=x^2+y^2+z^2-3;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,**$)%\"xG\"\"#\"\"\"\"\"\"*$)%
\"yGF)F*F+*$)%\"zGF)F*F+!\"$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "g:=x*y+y*z+z*x-3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,**
&%\"xG\"\"\"%\"yGF(F(*&F)\"\"\"%\"zGF(F(*&F,F+F'F+F(!\"$F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "h:=x^2*y+y^2*z+z^2*x-3;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"hG,**&)%\"xG\"\"#\"\"\"%\"yG\"\"\"F,*&)F
+F)F*%\"zGF,F,*&)F/F)F*F(F,F,!\"$F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 91 "Compute all solutions of f=g=h=0 using resultants. E-mail your \+
answer to hoeij@math.fsu.edu" }}{PARA 0 "" 0 "" {TEXT -1 95 "How many \+
different solutions (x,y,z) in C^3 are there? (here C stands for the c
omplex numbers)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "F:=x^4+
alpha*x^2+alpha*(x-alpha)-alpha*99/32;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"FG,**$)%\"xG\"\"%\"\"\"\"\"\"*&%&alphaGF+)F(\"\"#F*F+*&F-F*,
&F(F+F-!\"\"F+F+F-#!#**\"#K" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "C
ompute the square-free factorization of F for all possible values alph
a in C. Note that there are only finitely many alpha's for which F is \+
not square-free. E-mail your answer to hoeij@math.fsu.edu" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F1 := x^4+x+1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#F1G,(*$)%\"xG\"\"%\"\"\"\"\"\"F(F+F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "F2 := x^4+x^3+1;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#F2G,(*$)%\"xG\"\"%\"\"\"\"\"\"*$)F(\"\"$F*F+F+F+" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "If a1 is a root of F1 and a2 i
s a root of F2, then what are the possible minimum polynomials in Q[x]
 of a1-a2?" }}{PARA 0 "" 0 "" {TEXT -1 210 "Recall that the minimum po
lynomial of an algebraic number is the monic polynomial in Q[x] of min
imal degree that has that algebraic number as a root. And note that th
is minimum polynomial is always irreducible." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "E-mail me the minimum polynomia
l(s) that you found." }}}}{MARK "9" 0 }{VIEWOPTS 1 1 0 1 1 1803 }