 reserve K,F,E for Field,
         R,S for Ring;

theorem :: Kronecker
   for F being polynomial_disjoint 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 polynomial_disjoint 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
A1:  p is_a_irreducible_factor_of f by FIELD_1:3;
     reconsider p as irreducible Element of the carrier of Polynom-Ring F
     by A1,FIELD_1:def 1;
     consider q being Element of the carrier of Polynom-Ring F such that
A2:  p * q = f by A1,FIELD_1:def 1,GCD_1:def 1;
     consider E being FieldExtension of F such that
A3:  p is_with_roots_in E by Lm11;
     take E;
     consider a being Element of E such that
A4:  a is_a_root_of p,E by A3;
     reconsider p1 = p, q1 = q as Polynomial of F;
A5:  F is Subring of E by Def1;
     Ext_eval(f,a) = Ext_eval(p1*'q1,a) by A2,POLYNOM3:def 10
     .= Ext_eval(p1,a) * Ext_eval(q1,a) by A5,ALGNUM_1:20
     .= 0.E by A4; then
     a is_a_root_of f,E;
     hence thesis;
   end;
