
theorem pr1:
for F being Field,
    E being FieldExtension of F
for p being Element of the carrier of Polynom-Ring F
for a being Element of E
holds Ext_eval(p,a) = 0.E iff Ext_eval(NormPolynomial p,a) = 0.E
proof
let F be Field, E be FieldExtension of F;
let p be Element of the carrier of Polynom-Ring F;
let a be Element of E;
the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
     by FIELD_4:10;
then reconsider q = p as Element of the carrier of Polynom-Ring E;
reconsider qa = q as Polynomial of E;
reconsider ra = rpoly(1,a) as Element of the carrier of Polynom-Ring E
   by POLYNOM3:def 10;
A: now assume Ext_eval(p,a) = 0.E; then
   eval(qa,a) = 0.E by FIELD_4:26; then
   ra divides NormPolynomial qa by RING_4:26,RING_5:11; then
   eval(NormPolynomial q,a) = 0.E by RING_5:11;
   hence Ext_eval(NormPolynomial p,a) = 0.E by pr0,FIELD_4:26;
   end;
now assume Ext_eval(NormPolynomial p,a) = 0.E;
   then eval(NormPolynomial q,a) = 0.E by pr0,FIELD_4:26;
   then ra divides NormPolynomial qa by RING_5:11;
   then eval(qa,a) = 0.E by RING_5:11,RING_4:26;
   hence Ext_eval(p,a) = 0.E by FIELD_4:26;
   end;
hence thesis by A;
end;
