
theorem YY:
for F being Field
for E being FieldExtension of F
for p being non constant Element of the carrier of Polynom-Ring F
for q being non constant Element of the carrier of Polynom-Ring E
st p = q holds p is separable iff q is separable
proof
let F be Field, E be FieldExtension of F;
let p be non constant Element of the carrier of Polynom-Ring F;
let q be non constant Element of the carrier of Polynom-Ring E;
assume AS: p = q;
A: now assume p is separable; then
   B: p gcd (Deriv F).p = 1_.(F) by Thsepgcd;
   (Deriv E).q = (Deriv F).p by AS,FIELD_14:66; then
   q gcd (Deriv E).q = 1_.(F) by AS,B,FIELD_14:45 .= 1_.(E) by FIELD_4:14;
   hence q is separable by Thsepgcd;
   end;
now assume q is separable; then
   B: q gcd (Deriv E).q = 1_.(E) by Thsepgcd;
   (Deriv E).q = (Deriv F).p by AS,FIELD_14:66; then
   p gcd (Deriv F).p = 1_.(E) by AS,B,FIELD_14:45 .= 1_.(F) by FIELD_4:14;
   hence p is separable by Thsepgcd;
   end;
hence thesis by A;
end;
