
theorem mi2:
for F being Field
for p being irreducible Element of the carrier of Polynom-Ring F
ex E being F-finite FieldExtension of F
st deg(E,F) = deg p & p is_with_roots_in E
proof
let F be Field;
let p1 be irreducible Element of the carrier of Polynom-Ring F;
set p = NormPolynomial p1;
consider E being FieldExtension of F,
         a being F-algebraic Element of E such that
A: p = MinPoly(a,F) by mi1,RING_4:28;
reconsider L = FAdj(F,{a}) as F-finite FieldExtension of F;
take L;
p1 is non constant; then
len p1 - 1 > 0 by HURWITZ:def 2; then
B: len p1 <> 0;
deg p = len p - 1 by HURWITZ:def 2
     .= len p1 - 1 by B,POLYNOM5:57
     .= deg p1 by HURWITZ:def 2;
hence deg(L,F) = deg p1 by A,FIELD_6:67;
a in {a} & {a} is Subset of FAdj(F,{a}) by TARSKI:def 1,FIELD_6:35; then
reconsider a1 = a as Element of L;
L is Subring of E by FIELD_5:12; then
reconsider E as L-extending FieldExtension of F by FIELD_4:def 1;
for a being Element of E, b being Element of L st a = b
holds Ext_eval(p,a) = Ext_eval(p,b) by FIELD_7:14; then
Ext_eval(p,a1) = Ext_eval(p,a)
              .= 0.E by A,FIELD_6:52
              .= 0.L by EC_PF_1:def 1; then
Ext_eval(p1,a1) = 0.L by FIELD_6:25;
hence p1 is_with_roots_in L by FIELD_4:def 3,FIELD_4:def 2;
end;
