// Solution to Q2

// Ring Z5[x, y]
ring r = 5, (x, y), lp;

poly f1 = x^2 + y^2 + 1;
poly f2 = x^2*y + 2*x*y + x;

ideal J = f1, f2;

printf("Compute G = GB(f1, f2). If G ==1, then there are no solutions
in the closure.");

printf("If G !=1, then there are solutions, somewhere in the
closure");

ideal G;

G = groebner(J);

printf("GB G is:");
groebner(G);

printf("Note that since G has non-unit polynomials, there are
solutions over the closure");


printf("Now create the ideal of vanishing polyniomials in Z5: ideal J0
= x^5-x, y^5-y");

ideal J0 = x^5 -x, y^5-y;

printf("Compute G = GB(J+J0):");
G = groebner(J+J0);

G;