Dummit Foote Exc. 13.5.5
For any prime $p$ and any nonzero $a \in \mathbb{F}_p$ prove that $x^p -x +a$ is irreducible and separable over $\mathbb{F_p}$· [For the irreducibility: One approach - prove first that if $\alpha$ is a root then $\alpha + 1 $ is also a root. Another approach - suppose it's reducible and compute derivatives.]
Here is the answer for the first approach
I want to try second approach. Polynomial is clear separable by taking derivative we can see .
Let $x^p -x +a=f(x)g(x)$ then computing derivative gives $f(x)g'(x)+f'(x)g(x) =-1$.
So we have $f(x)$ and $g(x)$ and separable and $\gcd(f,g)=1$
this also tells me $f'(x)$ and $g'(x)$ are relatively prime and since $f(x)$ and $g(x)$ are and separable $\gcd(g,g')=1$
I only know this much in the second approach any hints to proceed furthur.
