Hey guys.

I have an exam tomorrow, and one of my practice problems asks me to find the splitting field (and its degree) of $\displaystyle x^5-4$ over $\displaystyle \mathbb{Q}$.

Clearly $\displaystyle u=4^{1/5}$ is a root, and

$\displaystyle x^5-4=(x-u)(x^4+ux^3+u^2x^2+u^3x+u^4)$.

Beyond that, though, I don't know where to go. I might just be missing something deceptively simple. Anyway, some assistance would be much appreciated!