
theorem
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 card Roots(E,p) = deg p
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;
   the carrier of Polynom-Ring F c= the carrier of Polynom-Ring K
     by FIELD_4:10; then
   reconsider q = p as Element of the carrier of Polynom-Ring K;
   deg p > 0 & deg p = deg q by RING_4:def 4,FIELD_4:20; then
   reconsider q as non constant Element of the carrier of Polynom-Ring K
     by RING_4:def 4;
   p splits_in K by FIELD_8:def 1; then
   consider c being non zero Element of K, r being Ppoly of K such that
   H: p = c * r by FIELD_4:def 5;
   now assume q is inseparable; then
     consider E being FieldExtension of K such that
     B: not for a being Element of E holds multiplicity(q,a) <= 1  by ThSep02;
     consider a being Element of E such that
     C: multiplicity(q,a) > 1 by B;
     reconsider E as K-extending FieldExtension of F;
     reconsider a as Element of E;
     multiplicity(q,a) = multiplicity(p,a) by sepsep;
     hence contradiction by C,AS,ThSep02;
     end; then
   D: card(Roots q) = deg q by H,FIELD_4:def 5,Thsepspl
                   .= deg p by FIELD_4:20;
   Roots q = Roots(K,p) by FIELD_7:13;
   hence ex E being FieldExtension of F st card Roots(E,p) = deg p by D;
   end;
now assume ex E being FieldExtension of F st card Roots(E,p) = deg p; then
   consider E being FieldExtension of F such that
   AS: card Roots(E,p) = deg p;
   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 q as Polynomial of E;
   deg p > 0 & deg p = deg q by RING_4:def 4,FIELD_4:20; then
   reconsider q as non constant Polynomial of E by RATFUNC1:def 2;
   A: card(Roots q) = card Roots(E,p) by FIELD_7:13
                   .= deg q by AS, FIELD_4:20;
   consider c being non zero Element of E, r being Ppoly of E such that
   H: q = c * r by FIELD_4:def 5,A,ThsepsplA;
   now let a be Element of E;
     multiplicity(q,a) = multiplicity(p,a) by FIELD_14:def 6;
     hence multiplicity(p,a) <= 1 by A,ThsepsplA;
     end;
   hence p is separable by H,FIELD_4:def 5,ThSep03;
   end;
hence thesis by A;
end;
