
theorem minpzeta:
MinPoly(zeta,F_Rat) = X^2+X+1
proof
H: Roots(F_Complex,X^2+X+1) =
   {a where a is Element of F_Complex : a is_a_root_of X^2+X+1,F_Complex}
   by FIELD_4:def 4;
zeta in Roots(F_Complex,X^2+X+1) by rootz,TARSKI:def 2;
then consider a being Element of F_Complex such that
I: a = zeta & a is_a_root_of X^2+X+1,F_Complex by H;
Ext_eval(X^2+X+1,zeta) = 0.F_Complex by I,FIELD_4:def 2;
hence thesis by FIELD_6:52;
end;
