
theorem
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E
holds deg(FAdj(F,{a}),F) = deg MinPoly(a,F)
proof
let F be Field, E be FieldExtension of F; let a be F-algebraic Element of E;
set B = Base a, m = deg MinPoly(a,F);
B: card B = m by lembascard;
C: B is Basis of VecSp(FAdj(F,{a}),F) by lembas; then
A: VecSp(FAdj(F,{a}),F) is finite-dimensional by MATRLIN:def 1;
then dim VecSp(FAdj(F,{a}),F) = deg MinPoly(a,F) by C,B,VECTSP_9:def 1;
hence thesis by A,FIELD_4:def 7;
end;
