
theorem
for p being Prime
for n being non zero Nat
ex F being finite Field st Char F = p & order F = p|^n
proof
let p be Prime, n be non zero Nat;
set E = the SplittingField of X^(p|^n,PrimeField(Z/p));
set F = InducedSubfield Roots(E,X^(p|^n,PrimeField(Z/p)));
A: the carrier of F = Roots(E,X^(p|^n,PrimeField(Z/p))) by dis;
reconsider F as finite Field;
take F;
thus Char F = p by RING_3:def 6;
thus thesis by A,lemex;
end;
