
theorem split:
for p being Prime
for n being non zero Nat
for F being Field st card F = p|^n
holds F is SplittingField of X^(p|^n,PrimeField F)
proof
let p be Prime, n be non zero Nat, F be Field;
assume AS: card F = p|^n;
Char F = p by T5,AS; then
F is p-characteristic & F is finite by AS,RING_3:def 6; then
reconsider U = PrimeField F as p-characteristic finite Field;
reconsider F as FieldExtension of U by FIELD_4:7;
reconsider q = X^(p|^n,PrimeField F) as
        non constant Element of the carrier of Polynom-Ring U;
B: Roots(F,q) = the carrier of F
   proof
   H: Roots(F,q) = {a where a is Element of F : a is_a_root_of q,F}
      by FIELD_4:def 4;
   now let o be object;
     assume o in the carrier of F; then
     reconsider a = o as Element of F;
     Ext_eval(X^(p|^n,U),a) = 0.F by AS,thX1e;
     then a is_a_root_of q,F by FIELD_4:def 2;
     hence o in Roots(F,q) by H;
     end;
   then the carrier of F c= Roots(F,q);
   hence thesis by XBOOLE_0:def 10;
   end; then
D: card Roots(F,q) = deg q by AS,Lm12; then
C: q splits_in F by lemMA;
   now let E be FieldExtension of U;
   assume D: q splits_in E & E is Subfield of F; then
   F is E-extending FieldExtension of U by FIELD_4:7; then
   E: Roots(F,q) c= Roots(E,q) by C,D,FIELD_8:28;
   Roots(E,q) c= the carrier of E; then
   F: the carrier of F c= the carrier of E by B,E;
   the carrier of E c= the carrier of F by D,EC_PF_1:def 1; then
   G: the carrier of F = the carrier of E by F,XBOOLE_0:def 10;
      the addF of E = (the addF of F) || the carrier of E &
      the multF of E = (the multF of F) || the carrier of E &
      1.E = 1.F & 0.E = 0.F by D,EC_PF_1:def 1;
   hence E == F by G;
   end;
hence thesis by lemMA,D,FIELD_8:def 1;
end;
