Let $F$ be a field of characteristic $p$.
Show that if $x^p-x-a$ is reducible in $F[x]$, then it splits into distinct factors in $F[x]$.
I have done the following:
We want to show that $x^p-x-a=f(x) \cdot g(x), f(x), g(x) \in F[x]$.
Let $h(x)=x^p-x-a \Rightarrow h'(x)=px^{p-1}-1=-1$, since $char F=p$
Since $h'(x) \not \equiv 0$, we have that $(h, h')=1$, that means that $h(x)$ is separable, so it hasn't multiple roots, so $f(x) \not \equiv g(x)$.
Is this correct??
I think that the problem is incorrectly stated. The fact is that if $X^p-X-a$ is reducible over $F$, then it splits into linear factors, and this is surely the intent of the question given to you. You prove this by observing that if $\theta$ is a root of one factor and $\lambda$ is a root of another factor, then there is an integer $m$ such that $\lambda=\theta+m$. The rest I’m sure you can do.