reserve i,j for Nat;
reserve A,B for Ring;
reserve K, L for Field;

theorem Th40:
  for K,L be Field, z be Element of L st K is Subring of L &
  z is_integral_over K
  ex f be Element of Polynom-Ring K
  st f <> 0_.K & {f}-Ideal = Ann_Poly(z,K) & f = NormPolynomial(f)
  proof
    let K,L be Field;
    let z be Element of L;
    assume that
A0: K is Subring of L and
A1: z is_integral_over K;
    consider f be Element of Polynom-Ring K such that
A2: {f}-Ideal = Ann_Poly(z,K) by A0,Th34;
A3: f <> 0.Polynom-Ring K by A1,A2,Th35,IDEAL_1:47;
    reconsider f as Element of Polynom-Ring K;
A4: f <> 0_.K by A3,POLYNOM3:def 10;
    set g = NormPolynomial(f);
A7: {g}-Ideal = Ann_Poly(z,K) by A2,RING_4:27,RING_2:21;
    g <> 0.Polynom-Ring K by A1,A7,Th35,IDEAL_1:47; then
A8: g <> 0_.K by POLYNOM3:def 10; then
A9: g is non zero Element of the carrier of Polynom-Ring K by Lm37;
A10:f is non zero Element of the carrier of Polynom-Ring K by A4,Lm37;
    g = NormPolynomial(g) by A9,A10,RING_4:24;
    hence thesis by A7,A8;
  end;
