Let $K$ be a field and $m$ a positive integer, not divisible by the characteristic of $K$. Let $L$ be the splitting field of the polynomial $X^{m}-1$ over $K$. Show that $\operatorname{Gal}(L / K)$ is isomorphic to a subgroup of $(\mathbb{Z} / m \mathbb{Z})^{*}$.

Now assume that $K$ is a finite field with $q$ elements. Show that $[L: K]$ is equal to the order of the residue class of $q$ in the group $(\mathbb{Z} / m \mathbb{Z})^{*}$. Hence or otherwise show that the splitting field of $X^{11}-1$ over $\mathbb{F}_{4}$ has degree 5 .