
theorem
for F being ordered Field,
    P being Ordering of F
for p being irreducible Element of the carrier of Polynom-Ring F
for E being FieldExtension of F,
    a being Element of E
st deg p is odd & a is_a_root_of p,E holds P extends_to FAdj(F,{a})
proof
let F be ordered Field, P being Ordering of F,
    p be irreducible Element of the carrier of Polynom-Ring F;
let E be FieldExtension of F, a be Element of E;
H: deg NormPolynomial p = deg p by lemdeg;
assume AS: deg p is odd & a is_a_root_of p,E;
then A: Ext_eval(p,a) = 0.E by FIELD_4:def 2;
then reconsider a as F-algebraic Element of E by FIELD_6:43;
MinPoly(a,F) = NormPolynomial p by A,FIELD_8:15;
then deg(FAdj(F,{a}),F) = deg(NormPolynomial p) by FIELD_6:67;
hence thesis by AS,H,main;
end;
