Let $F$ be a field of characteristic $p$, and consider the polynomial $f(x) = x^p -a$ in $F[x]$. I want to show that $f(x)$ is either irreducible or splits over $F$. It's easy to show that if there is a root, then $f(x)$ splits. If $f(x)$ does not have a root, then it cannot split, so I should be able to show that if $f(x)$ is reducible, then it must have a root.
I found a hint by this earlier post and was able to continue in the following manner:
Suppose $f(x)$ is reducible. Then we can write $f(x) = g(x) h(x)$. Let $E$ be the splitting field for the polynomial $g(x)$. We note in this field that $g(x)$ and therefore $f(x)$ has a root, $\beta$ and so we can write $f(x) = (x-\beta)^p$.
This means $g(x) = (x-\beta)^k$. Since all of the coefficients are over $F$, we must have that $k\beta \in F$. However, I don't see how this can show that $\beta$ is in $F$.