1.) F is a field of prime characteristic p. Prove that K = {x in F for all x^p=x} is a subfield of F.

For the subfield test, I showed that a-b is in K whenever a and b are in K, but I'm getting stuck when I try to show that ab inverse is in K.

2.) Show that any finite field has order p^n, where p is prime.

I have that Char F = p and the order of a divides p, and I know that the order of a is either 1 or p. I also know that p divides the order of the field, so F must contain an element of order p. I'm just having trouble connecting that to proving any finite field has order p^n.

Any help would be greatly appreciated. Thanks.