$t^p - t -a $ is irreducible if and only if it has no root

Question

Let $F$ be a field with characteristic $p$, and $f = t^p -t -a \in F[t]$. Show that $f$ is irreducible if and only if it has no root in $F$.

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The "only if" part is clear. But how do I show the "if" part? Any help is appreciated.