
theorem
for F being Field
for p being non constant Element of the carrier of Polynom-Ring F
ex E being F-finite FieldExtension of F
st p is_with_roots_in E & deg(E,F) <= deg p
proof
let F be Field, p be non constant Element of the carrier of Polynom-Ring F;
consider q being Element of Polynom-Ring F such that
A: q is_a_irreducible_factor_of p by FIELD_1:3;
reconsider q as Element of the carrier of Polynom-Ring F;
consider E being F-finite FieldExtension of F such that
C: deg(E,F) = deg q & q is_with_roots_in E by A,mi2;
take E;
reconsider p1 = p, q1 = q as Polynomial of F;
q1 divides p1 by A; then
consider r1 being Polynomial of F such that
D: q1 *' r1 = p1 by RING_4:1;
consider a being Element of E such that
E: a is_a_root_of q,E by C,FIELD_4:def 3;
F is Subring of E by FIELD_4:def 1; then
Ext_eval(p1,a) = Ext_eval(q1,a) * Ext_eval(r1,a) by D,ALGNUM_1:20
              .= 0.E * Ext_eval(r1,a) by E,FIELD_4:def 2;
hence p is_with_roots_in E by FIELD_4:def 3,FIELD_4:def 2;
q1 divides p1 by A;
hence deg(E,F) <= deg p by C,RING_5:13;
end;
