
theorem lemex:
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 card Roots(E,X^(p|^n,PrimeField F)) = p|^n
proof
let p be Prime, n be non zero Nat, F be p-characteristic Field;
let E be SplittingField of X^(p|^n,PrimeField F);
X^(p|^n,PrimeField F) splits_in E by FIELD_8:def 1;
then card Roots(E,X^(p|^n,PrimeField F)) = deg X^(p|^n,PrimeField F) by lemMA;
hence thesis by Lm12;
end;
