Let $\mathbb{k}$ be the field of all the $n$th roots of $1$ and $\mathbb{F=k}(\alpha_1, ..., \alpha_n)$ is a Kummer's field over $\mathbb{k}$, where $\alpha^n_i=a_i \in \mathbb{k^*}$, $i=1,\dots,m$.
Find isomorphism $Gal(\mathbb{F/k})\cong ((\mathbb{k^*})^n,a_1,\dots,a_m)/(\mathbb{k^*})^n$,
where $(\mathbb{k^*})^n=\{a^n | a \in \mathbb{k^*} \}$ is a subgroup of $n$-th exponents of elements of group $\mathbb{k^*}$.
I have no idea how to deal with Kummer's field since I'm absolutely new to Galois theory. And I would appreciate any hints on this task. Thanks in advance!