
theorem mmmz:
MinPoly(zeta,FAdj(F_Rat,{2-CRoot(2)})) = X^2+X+1
proof
set K = FAdj(F_Rat,{2-CRoot(2)});
    K = FAdj(F_Rat,{2-Root(2)}) by mmk; then
K: the carrier of K c= the carrier of F_Real by EC_PF_1:def 1;
   the carrier of Polynom-Ring F_Rat c=
                    the carrier of Polynom-Ring K by FIELD_4:10; then
reconsider p = X^2+X+1 as Element of the carrier of Polynom-Ring K;
M: deg p = 2 by LL,FIELD_4:20; then
reconsider p as non constant Element of the carrier of Polynom-Ring K
   by RING_4:def 4;
L: F_Rat is Subfield of K by FIELD_4:7;
LC p = LC X^2+X+1 by FIELD_8:5
    .= 1.F_Rat by RATFUNC1:def 7
    .= 1.K by L,EC_PF_1:def 1; then
A: p is monic by RATFUNC1:def 7;
B: now assume p is reducible;
   then p is with_roots by M,thirred2; then
   consider a being Element of K such that
   C: a is_a_root_of p by POLYNOM5:def 8;
   Y: Roots p c= Roots(F_Complex,p) by FIELD_4:28;
   Z: Roots(F_Complex,p) = {zeta, zeta^2} by rootz,mmo;
   a in Roots p by C,POLYNOM5:def 10; then
   per cases by Z,Y,TARSKI:def 2;
   suppose a = zeta;
     hence contradiction by lemlem,K;
     end;
   suppose a = zeta^2;
     hence contradiction by lemlem,K;
     end;
   end;
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;
0.F_Complex = Ext_eval(X^2+X+1,zeta) by I,FIELD_4:def 2
           .= Ext_eval(p,zeta) by FIELD_8:6;
hence thesis by A,B,FIELD_6:52;
end;
