
theorem lemsep3:
for F being Field
for p being non zero 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 p gcd (Deriv F).p <> 1_.(F)
proof
let F be Field, p be non zero 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;
  p <> 0_.(F); then p1 <> 0_.(E) by FIELD_4:12; then
reconsider p1 as non zero Element of the carrier of Polynom-Ring E
   by UPROOTS:def 5;
set g = p1 gcd (Deriv E).p1;
A: now assume ex a being Element of E st multiplicity(p,a) > 1; then
   consider a being Element of E such that A1: multiplicity(p,a) > 1;
   A6: multiplicity(p1,a) = multiplicity(p,a) by FIELD_14:def 6; then
   A2: a is_a_root_of p1 by A1,UPROOTS:52;
       eval(p1,a) = 0.E by A6,A1,UPROOTS:52,POLYNOM5:def 7; then
   A3: rpoly(1,a) divides p1 by RING_5:11;
       eval((Deriv E).p1,a) = 0.E by A1,A2,A6,multi4; then
       rpoly(1,a) divides (Deriv E).p1 by RING_5:11; then
   A4: deg rpoly(1,a) <= deg(p1 gcd (Deriv E).p1) by A3,RING_4:52,RING_5:13;
   A5: deg rpoly(1,a) = 1 by HURWITZ:27;
       1_.E = (1.E)|E & 0.E <> 1.E by RING_4:14; then
   A7: p1 gcd (Deriv E).p1 <> 1_.(E) by A4,A5,RING_4:21;
   (Deriv E).p1 = (Deriv F).p & 1_.(F) = 1_.(E) by FIELD_14:66,FIELD_4:14;
   hence p gcd (Deriv F).p <> 1_.(F) by A7,FIELD_14:45;
   end;
now assume B1: p gcd (Deriv F).p <> 1_.(F);
   (Deriv E).p1 = (Deriv F).p & 1_.(F)=1_.(E) by FIELD_14:66,FIELD_4:14; then
   p1 gcd (Deriv E).p1 <> 1_.(E) by B1,FIELD_14:45; then
   B8: g divides p1 & g divides (Deriv E).p1 & deg g > 0 by mm0,RING_4:52; then
   reconsider g as non constant Polynomial of E by RATFUNC1:def 2;
   consider r being Polynomial of E such that
   B2: p1 = g *' r by RING_4:52,RING_4:1;
   consider b being non zero Element of E, q being Ppoly of E such that
   M: p = b * q by AS,FIELD_4:def 5;
   B10: p1 splits_in E by M,FIELD_4:def 5;
   B9: E is FieldExtension of E by FIELD_4:6;
   r is non zero & g is non constant Polynomial of E by B2; then
   consider b being non zero Element of E, q being Ppoly of E such that
   B5: g = b * q by B2,B10,B9,FIELD_13:7,FIELD_4:def 5;
   consider a being Element of E such that
   B3: a is_a_root_of q by POLYNOM5:def 8;
       eval(b*q,a) = b * 0.E by B3,RING_5:7; then
   B7: a is_a_root_of g by B5; then
       a is_a_root_of p1 by mm1,RING_4:52; then
   B9: multiplicity(p1,a) > 1 iff eval((Deriv E).p1,a) = 0.E by multi4;
   multiplicity(p1,a) = multiplicity(p,a) by FIELD_14:def 6;
   hence ex a being Element of E st multiplicity(p,a) > 1
           by B7,B8,B9,POLYNOM5:def 7,mm1;
   end;
hence thesis by A;
end;
