Solving an equation in field $\mathbb Z_{11}$ using GAP

Question

Here is a solved question. I want for myself to examine it with GAP. So I did the following:

 > t:=Field( Z(11) );;
 > s:=Elements(t);

 [ 0*Z(11), Z(11)^0, Z(11), Z(11)^2, Z(11)^3, Z(11)^4, Z(11)^5, Z(11)^6, 
   Z(11)^7, Z(11)^8, Z(11)^9 ]

 > for alpha in s do
     if alpha^(12)-alpha^(10)= s[3] then Print(alpha,"\n");
     fi;
   od;

The final answers are:

   Z(11)^4,  Z(11)^9

Is this attempt acceptable from your point of view? Any other suggestions will be appreciated. (-:

Answer 1

To find a list of roots of a univariate polynomial, use RootsOfUPol (see ?RootsOfUPol in GAP for the documentation):

gap> x:=Indeterminate(GF(11),"x");
x
gap> f:=x^12-x^10-Z(11);
x^12-x^10+Z(11)^6
gap> RootsOfUPol(f);
[ Z(11)^9, Z(11)^4 ]

Note that these two are precisely 6 and 5 modulo 11, respectively:

gap> List(last,Int);
[ 6, 5 ]