Showing that $X^p-X-a$ is irreducible over a field of characteristic $p$ unless $a = b^p-b$

Question

Let $F$ be a field with characteristic $p$. In Milne's Fields and Galois Theory (page 72), it is said that the polynomial $X^p - X - a$ is irreducible in $F[X]$ unless $a = b^p - b$ for some $b \in F$.

In the latter case, $b$ would be a root of $X^p - X - a$, so it must be reducible.

Now I am working with the case where there is no such $b$. However, I really don't know how to approach this problem in the first place. For instance, I am not sure how the roots of this polynomial would look like over its splitting field.

Could you please help me with this problem? Thank you!