
theorem main: :: Kronecker
for F being Field
for f being non constant Element of the carrier of Polynom-Ring F
ex E being FieldExtension of F st f is_with_roots_in E
proof
let F be Field;
let f be non constant Element of the carrier of Polynom-Ring F;
consider p being Element of the carrier of Polynom-Ring F such that
A: p is_a_irreducible_factor_of f by FIELD_1:3;
reconsider p as irreducible Element of the carrier of Polynom-Ring F by A;
consider q being Element of the carrier of Polynom-Ring F such that
B: p * q = f by A,GCD_1:def 1;
consider E being FieldExtension of F such that
C: p is_with_roots_in E by kron;
take E;
consider a being Element of E such that
D: a is_a_root_of p,E by C,FIELD_4:def 3;
reconsider p1 = p, q1 = q as Polynomial of F;
F: F is Subring of E by FIELD_4:def 1;
Ext_eval(f,a) = Ext_eval(p1*'q1,a) by B,POLYNOM3:def 10
             .= Ext_eval(p1,a) * Ext_eval(q1,a) by F,ALGNUM_1:20
             .= 0.E * Ext_eval(q1,a) by D,FIELD_4:def 2;
hence thesis by FIELD_4:def 2,def 3;
end;
