I need to prove that if $f(x) = x^p - x - a$ has no root in $F$ then $f(x)$ is irreducible in $F[x]$, where $F$ is a field with characteristic $p$.
My attempt so far:
Let $L$ be a spliting field of $f(x)$ over $F$ and $u$ be a root of $f(x)$ in $L$. In other words, $u^p - u - a = 0$.
Assume that $f(x)$ is reducible in $F[x]$, which means $f(x) = g(x).h(x)$, where $\deg g \ge 1$, $\deg h \ge 1$.
This implies $x^p - x - (u^p - u) = g(x).h(x)$ (I substite $a$ by $u^p - u$)
Or $(x - u)^p - (x - u) = g(x).h(x)$ (in $L$ we have $(a+b)^p = a^p + b^p$)
I'm not sure how to continue from this point. My goal is to somehow prove that $u$ must belong to $F$, which concludes the proof.
Please give me a hint. Thank you.