I'm afraid I don't really get why this hint works, although I speant a lot of thought on it. Here's what I'm thinking:
For the first part of the question:
I can prove that the size of the field must be p^n for some n. If n = 1, then we could use Fermat's little theorem and the fact that the field must be isomorphic to Zp to show that the only root is 1. But firstly this only deals with the case n=1 (we would need to show that the field is isomorphic to "something else" to get the general case to work, but I'm not sure if this is a good idea). Secondly I'd much rather not appeal to Fermat's little theorem anyway. I'm sure there must be a nicer way to do this relying just on standard field theory.
For the second part of the question, I've seen the Frobenius automorphism and it looks as though it may be applicable here, though I cannot see how.
Also, I was aware of the results given in the hint in the post above, but really cannot see how it helps (though it does seem to point in the direction of the Frobenius automorphism).
Suppose we are working in , and we want to compute . If we use the binomial theorem we get . Just remember when we write we mean . Now the characheristic of is therefore and so .
Now, if is a prime and then divides .
Let where and so .
This means .
But if then all the terms become zero, so, .
Thus, .
And,
In general, .
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The mapping by is an (Frobenius) automorphism because and by above. It remains to show that is a bijection. However, since is finite it means all we have to show is that is an injection by the pigeonhole principle. Thus, say .
OK...I think I'm getting somewhere now. We can use your hint to show that the only p-th root of 1 in a field of characteristic p is 1.
Using the same idea, it seems like we can show that the only root of in a field of order q is 1, since it is a standard result that .
So it seems to me that factorises only as far as . Does this seem right??
EDIT: Actually I thought this looked a bit too easy to be true. I forgot that just because a polynomial doesn't have a root in a field K, it can still factorise as a product of two (or more) irreducibles. So I'm still quite stuck (and probably more confused than before...) - how do we factorise this?
EDIT 2: Sorry. I've just realised how silly I've been! Thanks for the help!