
theorem mpol3:
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 & Ext_eval(p,a) = 0.E)
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), g = hom_Ext_eval(a,F);
X: F is Subring of E by FIELD_4:def 1;
a is_integral_over F by alg1; then
m <> 0_.F & {m}-Ideal = Ann_Poly(a,F) & m = NormPolynomial(m)
       by X,ALGNUM_1:def 9; then
m in {p where p is Polynomial of F: Ext_eval(p,a) = 0.E} by IDEAL_1:66;
then consider m1 being Polynomial of F such that
Z1: m1 = m & Ext_eval(m1,a) = 0.E;
now assume A: p is monic & p is irreducible & Ext_eval(p,a) = 0.E;
  reconsider p1 = p as Element of Polynom-Ring F;
  ker g is principal by IDEAL_1:def 28; then
  consider u being Element of Polynom-Ring F such that
  C1: ker g = {u}-Ideal;
  hom_Ext_eval(a,F).p = 0.E by A,ALGNUM_1:def 11; then
  p in {v where v is Element of Polynom-Ring F : g.v = 0.E}; then
  C2: p in {u}-Ideal by C1,VECTSP10:def 9;
  p in the set of all u*r where r is Element of Polynom-Ring F
      by C2,IDEAL_1:64; then
  consider v being Element of Polynom-Ring F such that
  C3: p = u * v;
  reconsider u1 = u as Polynomial of F by POLYNOM3:def 10;
  A2: now assume u is Unit of Polynom-Ring F;
      then {u}-Ideal = [#](Polynom-Ring F) by RING_2:20;
      hence contradiction by C1;
      end;
  u divides p by C3; then
  u is_associated_to p by A2,A,RING_2:def 9;
  then {p}-Ideal = ker g by C1,RING_2:21;
  hence p = MinPoly(a,F) by A,mpol1;
  end;
hence thesis by Z1;
end;
