
theorem
for F being Field,
    E being FieldExtension of F
for p being irreducible Element of the carrier of Polynom-Ring F
for a,b being Element of E
st a is_a_root_of p,E & b is_a_root_of p,E
holds FAdj(F,{a}),FAdj(F,{b}) are_isomorphic
proof
let F1 be Field, E1 be FieldExtension of F1;
let p be irreducible Element of the carrier of Polynom-Ring F1;
let a,b be Element of E1;
assume a is_a_root_of p,E1 & b is_a_root_of p,E1; then
B: Ext_eval(p,a) = 0.E1 & Ext_eval(p,b) = 0.E1 by FIELD_4:def 2; then
reconsider a1 = a as F1-algebraic Element of E1 by FIELD_6:43;
id F1 is isomorphism; then
reconsider F2 = F1 as F1-isomorphic Field by RING_3:def 4;
reconsider E2 = E1 as FieldExtension of F2;
reconsider h = id F1 as Isomorphism of F1,F2;
reconsider b1 = b as F2-algebraic Element of E2 by B,FIELD_6:43;
now let i be Element of NAT;
  thus ((PolyHom h).p).i = h.(p.i) by FIELD_1:def 2 .= p.i;
  end; then
(PolyHom h).p = p; then
Psi(a1,b1,h,p) is h-extending isomorphism by B,unique1;
hence thesis;
end;
