
theorem lemMA:
for F being Field
for E being FieldExtension of F
for p being non constant Element of the carrier of Polynom-Ring F
holds card Roots(E,p) = deg p iff (p splits_in E & p is separable)
proof
let F be Field, E be FieldExtension of F;
let p be non constant Element of the carrier of Polynom-Ring F;
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;
H: deg p > 0 & deg q = deg p by RING_4:def 4,FIELD_4:20; then
reconsider q as non constant Polynomial of E by RATFUNC1:def 2;
A: now assume B: card Roots(E,p) = deg p;
   Roots q = Roots(E,p) by FIELD_7:13; then
   consider x being non zero Element of E, v being Ppoly of E such that
   C: q = x * v by H,B,FIELD_15:69,FIELD_4:def 5;
   thus p splits_in E & p is separable by B,C,FIELD_4:def 5,FIELD_15:85;
   end;
now assume AS: p splits_in E & p is separable;
  consider x being non zero Element of E, v being Ppoly of E such that
  A: p = x * v by AS,FIELD_4:def 5;
  now let a be Element of E;
    per cases;
    suppose C: not a is_a_root_of p,E;
      now assume a is_a_root_of q;
        then 0.E = Ext_eval(p,a) by FIELD_4:26;
        hence contradiction by C,FIELD_4:def 2;
        end;
      hence multiplicity(q,a) <= 1 by UPROOTS:52;
      end;
    suppose a is_a_root_of p,E; then
      multiplicity(p,a) = 1 by AS,FIELD_15:75;
      hence multiplicity(q,a) <= 1 by FIELD_14:def 6;
      end;
    end;
  hence deg p = card Roots q by H,A,FIELD_4:def 5,FIELD_15:69
             .= card Roots(E,p) by FIELD_7:13;
  end;
hence thesis by A;
end;
