
theorem x1000:
for F1 being Field,
    F2 being F1-isomorphic F1-homomorphic Field
for h being Isomorphism of F1,F2
for E1 being FieldExtension of F1, E2 being FieldExtension of F2
for a being F1-algebraic Element of E1, b being F2-algebraic Element of E2
for p being irreducible Element of the carrier of Polynom-Ring F1
st Ext_eval(p,a) = 0.E1 & Ext_eval((PolyHom h).p,b) = 0.E2
holds (PolyHom h).MinPoly(a,F1) = MinPoly(b,F2)
proof
let F1 be Field, F2 be F1-isomorphic F1-homomorphic Field;
let h be Isomorphism of F1,F2;
let E1 be FieldExtension of F1, E2 being FieldExtension of F2;
let a be F1-algebraic Element of E1, b being F2-algebraic Element of E2;
let p be irreducible Element of the carrier of Polynom-Ring F1;
assume AS: Ext_eval(p,a) = 0.E1 & Ext_eval((PolyHom h).p,b) = 0.E2;
hence MinPoly(b,F2) = NormPolynomial (PolyHom h).p by uu4
                   .= (PolyHom h).(NormPolynomial p) by uu5
                   .= (PolyHom h).MinPoly(a,F1) by AS,uu4;
end;
