 Solution

Every $\tau \in \mathrm{Aut}(L({\xi }_N)/K({\xi }_N))$ fixes $K$ and permutes the roots of $f$, so it preserves their splitting field $L$. Restriction therefore defines a homomorphismIf $\rho (\tau )$ 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: