Factor x^16-x over the fields F4 and F8
factored over Z (or Q), x^16-x = (x*(x - 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)
The Attempt at a Solution
I know the that quadratic and higher terms I have left do not have any linear factors.. I'm obviously missing something. But even with that I'm not really sure how to start checking- I *think* I'm right in saying I can check for irreducibility over F2 first, because that would be implied by irreducibility over the larger field. Then there are less polynomials to check, but still a fair few for the term of degree 8..
Thanks in advance :)