
theorem mpol1:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E,
    p being Element of the carrier of Polynom-Ring F
holds p = MinPoly(a,F) iff
      (p is monic & p is irreducible & ker(hom_Ext_eval(a,F)) = {p}-Ideal)
proof
let F be Field, E be FieldExtension of F;
let a be F-algebraic Element of E,
    p be Element of the carrier of Polynom-Ring F;
set m = MinPoly(a,F);
X: F is Subring of E by FIELD_4:def 1;
Y: a is_integral_over F by alg1; then
Z: m <> 0_.F & {m}-Ideal = Ann_Poly(a,F) & m = NormPolynomial(m)
       by X,ALGNUM_1:def 9;
now assume A0: p is monic & p is irreducible &
               ker(hom_Ext_eval(a,F)) = {p}-Ideal;
  then A1: p <> 0_.F;
  A2: {p}-Ideal = Ann_Poly(a,F) by A0,alg0;
  p = NormPolynomial(p) by A0,RING_4:24;
  hence p = m by X,Y,A1,A2,ALGNUM_1:def 9;
  end;
hence thesis by Z,alg0;
end;
