
theorem lemsep1:
for F being Field
for p being irreducible Element of the carrier of Polynom-Ring F
for E being FieldExtension of F st p splits_in E holds
(ex a being Element of E st multiplicity(p,a) > 1) iff (Deriv F).p = 0_.(F)
proof
let F be Field, p be irreducible Element of the carrier of Polynom-Ring F;
let E be FieldExtension of F;
assume AS: p splits_in E;
  the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
    by FIELD_4:10; then
  reconsider p1 = p as Element of the carrier of Polynom-Ring E;
  deg p1 = deg p by FIELD_4:20; then
reconsider p1 as non zero Element of the carrier of Polynom-Ring E
   by RATFUNC1:def 2,RING_4:def 4;
set g = p gcd (Deriv F).p;
H: g is monic Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
A: now assume A2: ex a being Element of E st multiplicity(p,a) > 1;
   g = 1_.(F) or g = NormPolynomial p by RING_4:52,H,lemir; then
   A4: deg g = deg p by AS,A2,lemsep3,REALALG3:11;
   now assume (Deriv F).p <> 0_.(F); then
     (Deriv F).p is non zero by UPROOTS:def 5; then
     deg g <= deg((Deriv F).p) by RING_4:52,RING_5:13;
     hence contradiction by A4,FIELD_14:64;
     end;
   hence (Deriv F).p = 0_.(F);
   end;
now assume (Deriv F).p = 0_.(F); then
   B1: p gcd (Deriv F).p = NormPolynomial p by FIELD_14:46;
   deg(NormPolynomial p) = deg p by REALALG3:11; then
   p gcd (Deriv F).p <> 1_.(F) by B1,RATFUNC1:def 2,RING_4:def 4; then
   consider a being Element of E such that
   B2: multiplicity(p,a) > 1 by AS,lemsep3;
   thus ex a being Element of E st multiplicity(p,a) > 1 by B2;
   end;
hence thesis by A;
end;
