
theorem lemma:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E,
    b being Element of E holds
b in the carrier of FAdj(F,{a}) iff
ex p being Polynomial of F st deg p < deg MinPoly(a,F) & b = Ext_eval(p,a)
proof
let F be Field, E be FieldExtension of F;
let a be F-algebraic Element of E, b be Element of E;
H: E is (Polynom-Ring F)-homomorphic;
A: now assume ex p being Polynomial of F
              st deg p < deg MinPoly(a,F) & b = Ext_eval(p,a); then
   b in the set of all Ext_eval(p,a) where p is Polynomial of F;
   then b in the carrier of RAdj(F,{a}) by H,FIELD_6:45;
   hence b in the carrier of FAdj(F,{a}) by H,FIELD_6:56;
   end;
now assume b in the carrier of FAdj(F,{a});
   then b in the carrier of RAdj(F,{a}) by H,FIELD_6:56;
   then b in the set of all Ext_eval(p,a) where p is Polynomial of F
      by H,FIELD_6:45;
   then consider q being Polynomial of F such that
   E: b = Ext_eval(q,a);
   set ma = MinPoly(a,F), r = q mod ma;
   B: F is Subring of E by FIELD_4:def 1;
   C: q = (q div ma) *' ma + r by RING_4:4;
   D: deg r < deg ma by FIELD_5:16;
   Ext_eval(q,a)
     = Ext_eval((q div ma) *' ma,a) + Ext_eval(r,a) by C,B,ALGNUM_1:15
    .= (Ext_eval(q div ma,a) * Ext_eval(ma,a)) + Ext_eval(r,a) by B,ALGNUM_1:20
    .= (Ext_eval(q div ma,a) * 0.E) + Ext_eval(r,a) by FIELD_6:52
    .= Ext_eval(r,a);
   hence ex p being Polynomial of F
                    st deg p < deg MinPoly(a,F) & b = Ext_eval(p,a) by D,E;
   end;
hence thesis by A;
end;
