
theorem
for p being Prime
for n being non zero Nat
for F being p-characteristic Field
for E being SplittingField of X^(p|^n,PrimeField F)
holds E == InducedSubfield Roots(E,X^(p|^n,PrimeField F))
proof
let p being Prime, n be non zero Nat, F be p-characteristic Field;
let E be SplittingField of X^(p|^n,PrimeField F);
set K = InducedSubfield Roots(E,X^(p|^n,PrimeField F));
A: PrimeField E = PrimeField K by RING_3:94;
PrimeField F is Subfield of E by FIELD_4:7;
then PrimeField E = PrimeField(PrimeField F) by RING_3:94
                 .= PrimeField F by RING_3:95;
then reconsider K as FieldExtension of (PrimeField F) by A,FIELD_4:7;
reconsider E1 = E as K-extending FieldExtension of (PrimeField F)
   by FIELD_4:7;
A: X^(p|^n,PrimeField F) splits_in E1 by FIELD_8:def 1;
the carrier of K = Roots(E,X^(p|^n,PrimeField F)) by dis;
then X^(p|^n,PrimeField F) splits_in K by A,FIELD_8:27;
hence thesis by FIELD_8:def 1;
end;
