
theorem ID2a:
for F being Field
for E being FieldExtension of F
for a being F-algebraic Element of E
for f being F-fixing Automorphism of FAdj(F,{a})
holds f.a in Roots(FAdj(F,{a}),MinPoly(a,F))
proof
let F be Field, E be FieldExtension of F, a be F-algebraic Element of E;
let f be F-fixing Automorphism of FAdj(F,{a});
set p = MinPoly(a,F);
a in {a} & {a} is Subset of FAdj(F,{a}) by TARSKI:def 1,FIELD_6:35; then
reconsider a1 = a as Element of FAdj(F,{a});
H1: E is FAdj(F,{a})-extending &
    p is Element of the carrier of Polynom-Ring F by FIELD_4:7;
H2: Roots(FAdj(F,{a}),p) =
    {b where b is Element of FAdj(F,{a}) : b is_a_root_of p,FAdj(F,{a})}
       by FIELD_4:def 4;
0.FAdj(F,{a}) = 0.E by FIELD_6:def 6
    .= Ext_eval(p,a) by FIELD_6:52 .= Ext_eval(p,a1) by H1,FIELD_6:11; then
a1 is_a_root_of p,FAdj(F,{a}) by FIELD_4:def 2; then
a1 in Roots(FAdj(F,{a}),p) by H2;
hence thesis by FIELD_8:38;
end;
