a question about field theory and polynomials

Question

Hello all I was given this question in my field theory class on which I would certainly appreciate the help: I am given a field F of characteristic p ($ ch(F) > 0 $) and this polynomial

$ f(x) = x^p - x -a $ such that $ a \in F^* $

I am asked to show that if $ g,h \in F[x] $ are irreducible factors of the polynomial f then $ deg(g)=deg(h) $

All I was able to do was to show that the derivative of the polynomial is $ f'(x) = px^{p-1}-1 = -1$

because p is the characteristic and this necessarily shows that f and f' are coprime but I have no idea how to really proceed to the claim I am asked to prove. Any help would be appreciated. Thank you

Answer 1

I don't know if you already solved this, but I felt like I should explain for anyone in the future since I deleted my original comment.

To expand on my comment (since deleted) and my second comment, notice that if $\alpha$ is a root of $f$, then $\alpha+i$ is also a root for $i$ in the characteristic subfield, $\Bbb{F}_p$ since $(\alpha+i)^p-\alpha-i -a=\alpha^p-\alpha-a+i^p-i=0+i-i=0$ since $i\in \Bbb{F}_p$. Therefore if $f$ has a root in a field, then it has at least $p$ distinct roots. Since it is a polynomial of degree $p$ these are all the roots. Thus if the polynomial has one root in a field, it factors completely.

Then hint: What is the degree of the splitting field?

We now have that the splitting field of the polynomial has degree $d$ over the base field where $d$ is the degree of any irreducible factor of $f$, since adjoining the root of any irreducible factor gives the splitting field. This is what we wished to show.