
theorem
for F being Field,
    p being non constant monic Element of the carrier of Polynom-Ring F
holds p is separable iff
      for E being SplittingField of p holds p is Ppoly of E,Roots(E,p)
proof
let F be Field,
    p be non constant monic Element of the carrier of Polynom-Ring F;
A: now assume AS: p is separable;
   thus for E being SplittingField of p holds p is Ppoly of E,Roots(E,p)
     proof
     let E be SplittingField of p;
     consider a being Element of E, q being Ppoly of E,Roots(E,p) such that
     A: p = a * q by AS,lempp;
     H: F is Subfield of E by FIELD_4:7;
     a = a * 1.E
      .= a * (LC q) by RING_5:50
      .= LC(a * q) by RING_5:5
      .= LC p by A,FIELD_8:5
      .= 1.F by RATFUNC1:def 7
      .= 1.E by H,EC_PF_1:def 1;
     hence thesis by A;
     end;
   end;
now assume AS:
   for E being SplittingField of p holds p is Ppoly of E,Roots(E,p);
   now let E be SplittingField of p;
     reconsider q = p as Ppoly of E,Roots(E,p) by AS;
     1.E * q = q;
     hence ex a being Element of E,
              q being Ppoly of E,Roots(E,p) st p = a * q;
     end;
   hence p is separable by lempp;
   end;
hence thesis by A;
end;
