
theorem
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E
holds MinPoly(a,F) = rpoly(1,a) iff a in the carrier of F
proof
let F be Field, E be FieldExtension of F;
let a be F-algebraic Element of E;
set ma = MinPoly(a,F), g = hom_Ext_eval(a,F);
X: F is Subring of E by FIELD_4:def 1;
A: now assume a in the carrier of F;
   then reconsider a1 = a as Element of F;
   reconsider p = rpoly(1,a1) as Element of the carrier of Polynom-Ring F
       by POLYNOM3:def 10;
   A3: rpoly(1,a) = rpoly(1,a1) by FIELD_4:21;
   deg p = 1 by HURWITZ:27; then
   A2: p is irreducible by RING_4:42;
   Ext_eval(p,a) = eval(p,a1) by FIELD427
                .= a1 - a1 by HURWITZ:29
                .= 0.F by RLVECT_1:15
                .= 0.E by X,C0SP1:def 3;
   hence ma = rpoly(1,a) by A3,A2,mpol3;
   end;
now assume ma = rpoly(1,a);
  then reconsider p = rpoly(1,a) as Element of the carrier of Polynom-Ring F;
  -p.0 = -rpoly(1,a).0 by X,Th19
     .= --(power(E).(a,1+0)) by HURWITZ:25
     .= --(power(E).(a,0) * a) by GROUP_1:def 7
     .= --(1_E * a) by GROUP_1:def 7
     .= a;
  hence a in the carrier of F;
  end;
hence thesis by A;
end;
