
theorem Thsepspl:
for F being Field
for p being separable non constant Element of the carrier of Polynom-Ring F
holds deg p = card(Roots p) iff p splits_in F
proof
let F be Field;
let p be separable non constant Element of the carrier of Polynom-Ring F;
deg p > 0 by RING_4:def 4; then
K: p is non constant Polynomial of F by RATFUNC1:def 2;
reconsider E = F as FieldExtension of F by FIELD_4:6;
reconsider pE = p as non constant Element of the carrier of Polynom-Ring E;
now assume B: p splits_in F; then
   consider c being non zero Element of F, q being Ppoly of F such that
   C: p = c * q by FIELD_4:def 5;
   reconsider qE = q as Element of the carrier of Polynom-Ring E
       by POLYNOM3:def 10;
   H1: c * q = @(c,E) * qE by FIELD_7:def 4;
   now let a be Element of F;
     assume a is_a_root_of q; then
     C3: eval(qE,@(a,E)) = 0.E by FIELD_7:def 4;
     Ext_eval(p,@(a,E)) = eval(pE,@(a,E)) by FIELD_4:26
                       .= @(c,E) * eval(qE,@(a,E)) by C,H1,POLYNOM5:30
                       .= 0.E by C3;
     hence 1 = multiplicity(p,@(a,E)) by B,ThSep0,FIELD_4:def 2
            .= multiplicity(p,a) by multi3
            .= multiplicity(q,a) by C,lems;
     end; then
   D: q is Ppoly of F,(Roots q) by FIELD_14:30;
   thus deg p = deg q by C,RING_5:4
             .= card(Roots q) by D,RING_5:60 .= card(Roots p) by C,RING_5:19;
   end;
hence thesis by K,ThsepsplA;
end;
