
theorem ThSep03:
for F being Field,
    p being non constant Element of the carrier of Polynom-Ring F
holds p is separable iff
      (ex E being FieldExtension of F st p splits_in E &
          for a being Element of E holds multiplicity(p,a) <= 1)
proof
let F be Field, p be non constant 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;
   for a being Element of K holds multiplicity(p,a) <= 1 by AS,ThSep01;
   hence ex E being FieldExtension of F st p splits_in E &
   for a being Element of E holds multiplicity(p,a) <= 1 by B;
   end;
now assume ex E being FieldExtension of F st p splits_in E &
          for a being Element of E holds multiplicity(p,a) <= 1; then
  consider E being FieldExtension of F such that
  AS: p splits_in E &
      for a being Element of E holds multiplicity(p,a) <= 1;
  now let a be Element of E;
    assume a is_a_root_of p,E; then
    multiplicity(p,a) >= 1 & multiplicity(p,a) <= 1 by AS,mulzero;
    hence multiplicity(p,a) = 1 by XXREAL_0:1;
    end;
  hence p is separable by AS,ThSep1;
  end;
hence thesis by A;
end;
