Is $x^{p}+\alpha$ irreducible over $\mathbb{F}_{p}(\alpha)$ for transcendent $\alpha$?

Question

Let $\mathbb{F}_{p}(\alpha)/\mathbb{F}_{p}$ be a transcendent field extension and $f(x)=x^{p}+\alpha$. Then could it be that $f$ is irreducible over $\mathbb{F}_{p}(\alpha)$? And why or why not?

I assume that it is.