Let $L/K$ be a Galois extension of fields and let $\zeta_n$ be a primitive $n$th root of unity in some field extension of $L$, where $n$ is not divisible by the characteristic. Prove that $\mathrm{Aut}(L(\zeta_n)/K(\zeta_n))$ is a subgroup of $\mathrm{Aut}(L/K)$.
I'm unsure because I don't really know what is required. If $\sigma \in \mathrm{Aut}(L(\zeta_n)/K(\zeta_n))$, then $\sigma$ maps $\zeta_n$ to another primitive $n$th root of unity. Why must it restrict to an automorphism of $L$?
Thanks