Given a field $F$ containing all the roots of unity I'm trying to show that $f(x) = x^p - \alpha^p$ is irreducible over $F$ (where $\alpha$ is not in $F$).
It's clear that $f$ splits in $F(\alpha)$ and the other roots are $\alpha \omega^i$ where $\omega$ is a primitive pth root of unity. Hence $F(\alpha)$ is the splitting field of $f$ - now if we could show this was also a separable extension then the Galois group would be transitive and $f$ would be irreducible. However I'm not sure if this is possible - it doesn't work when $charF = p$ for example because the extension won't be separable.
I'm guessing I probably need to use the fact that $p$ is a prime somehow but I'm struggling to see how!
Thanks for any tips