Example 3

Let L be a cyclic extension of K of degree $n$; then the group $(L^n \cap K^*)/K^{*n}$ is cyclic of order $n$. If $a \in K^*$ is such that the residue class of $a$ mod $K^{*n}$ is a generator of this group, then L is K-isomorphic to $K[X]/(X^n - a)$ and the group $\mathrm{Gal}(L/K)$ consists of the $n$ automorphisms mapping $X$ to $\zeta X$, $\zeta \in \mu_n(K)$.