Galois group of a cyclotomic extension

Question

Do you know a condition on the field $k$ for that the injection of $\text{Gal}(k[\zeta_n]/k)$ in $(\Bbb Z/n\Bbb Z)^*$ is bijective ?

It is the case for $k = \mathbb{Q}$, but not for $k = \mathbb{R}$ for example.

Answer 1

This is the case if and only if the cycloctomic polynomial $\Phi_n$ is irreducible over $k$. This is hard to check if $k$ is an arbitrary field. For $k=\mathbb{Q}$ it is always true, and if $k=\mathbb{F}_q$ with $n,q$ coprime it is true if and only if $[q]$ is a generator of $(\mathbb{Z}/n)^*$. The same holds for $k=\mathbb{Q}_q$ by Hensel's Lemma.