
theorem main1:
for F being Field,
    p being non constant irreducible Element of the carrier of Polynom-Ring F
holds p is separable iff (Deriv F).p <> 0_.(F)
proof
let F be Field,
    p be non constant irreducible Element of the carrier of Polynom-Ring F;
A: now assume AS: p is separable;
   set K = the SplittingField of p;
   B: p splits_in K by FIELD_8:def 1;
   not ex a being Element of K st multiplicity(p,a) > 1 by AS,ThSep02;
   hence (Deriv F).p <> 0_.(F) by B,lemsep1;
   end;
now assume AS: (Deriv F).p <> 0_.(F);
  set K = the SplittingField of p;
  p splits_in K by FIELD_8:def 1; then
  for a being Element of K holds multiplicity(p,a) <= 1 by AS,lemsep1;
  hence p is separable by ThSep01;
  end;
hence thesis by A;
end;
