= Solution
Every $\tau\in\operatorname{Aut}(L(\xi_N)/K(\xi_N))$ fixes $K$ and permutes the roots of $f$, so it preserves their <splitting field> $L$. Restriction therefore defines a homomorphism
$$
\rho:\operatorname{Aut}(L(\xi_N)/K(\xi_N))
\longrightarrow\operatorname{Aut}(L/K).
$$
If $\rho(\tau)$ is the identity, then $\tau$ fixes both $L$ and $\xi_N$, which generate $L(\xi_N)$; hence $\tau$ is the identity. Thus $\rho$ is injective, and its image is a subgroup:
$$
\boxed{\operatorname{Aut}(L(\xi_N)/K(\xi_N))
\hookrightarrow\operatorname{Aut}(L/K)}.
$$
Back to article page
