Finding the Galois group of $x^n-a$.

Question

I was wondering if there is some general method to find the Galois group of $p(x)=x^n-a$ over $\mathbb{Q}$. I know that the splitting field of $x^n-1$ over $\mathbb{Q}$ is $\mathbb{Q} (\zeta)$ with Galois group $\mathbb{Z}_n^{\times}$ where $\zeta$ are the primitive roots of unity, but what about $p(x)=x^n-a$ generally? Can I use this knowledge to find the Galois group of $p(x)=x^n-a$? I know that the solutions of $p(x)=x^n-a$ are of the form $a^{1/n} \zeta^i$ for $i=0,1,...n-1$. I am mostly interested in the case where $a^{1/n}$ is irrational.