Splitting field of $X^n-a$

Show that the splitting field of $X^n-a$ over a field $K$ is $K(\alpha, \zeta_n)$, where $\alpha$ is a $n$-th root of $a$ and $\zeta_n$ is a primitive $n$-root of unity.

Notice that $\alpha \zeta_n^k$, $k=0,\ldots,n-1$ are the roots of the polynomial given. So, the splitting field will contain both $\alpha$, (k=0) case and $\zeta_n$. This means that the splitting field contains $K(\alpha,\zeta_n)$. Also, if the splitting field contains $\alpha$ and $\zeta_n$, it also contains all the roots of the polynomial(products of $\alpha$ and $\zeta_n$) which means that the splitting field is exactly $K(\alpha,\zeta_n)$.