
theorem z24:
Roots X^2+X+1 = {}
proof
set p = <%1.(Z/2), 1.(Z/2), 1.(Z/2)%>;
set a = the Element of Roots p;
now assume A: Roots p <> {};
  then a in Roots p;
  then reconsider a as Element of Z/2;
  B: a is_a_root_of p by A,POLYNOM5:def 10;
  per cases by cz2,TARSKI:def 2;
  suppose a = 0.(Z/2); then
    eval(p,a) = 1.(Z/2) + 1.(Z/2) * 0.(Z/2) + 1.(Z/2) * (0.(Z/2))^2
                by evalq
             .= 1.(Z/2);
    hence contradiction by B;
    end;
  suppose a = 1.(Z/2);then
    eval(p,a) = 1.(Z/2) + 1.(Z/2) * 1.(Z/2) + 1.(Z/2) * (1.(Z/2))^2
                by evalq
             .= 1.(Z/2) + 0.(Z/2) by FIELD_3:4;
    hence contradiction by B;
    end;
  end;
hence thesis;
end;
