$\zeta_q\in L^{G({\frak P})}\Leftrightarrow$ if $\sigma\in$ $G(\frak p$) then $\sigma(\zeta_q)=\zeta_q\Leftrightarrow$ if $\sigma$ fixes $\frak P$ then $\sigma$ fixes $\zeta_q$. What's next?
Suppose $L/k$ is a Galois extension with Galois group $G$. For $\frak P$ an ideal of $\cal O$$_L$, the $decomposition$ $group$ $G(\frak P$) is the set {$\sigma\in G|\sigma(\frak P)=\frak P$}. I am looking for roots of unity that appear in $L^{G({\frak P})}$.
Let $\zeta_q\in L$ denote the $q$-th primitive root of unity ($p$ either prime, or the power of a prime, not necessarily the one that lies over $\frak P$). Then the above chain of equivalences is true. What's next? I know that does not imply $\zeta_q\in\frak P$. Does that tell us anything about $\cal O$$_L/\frak P$? I am having trouble making sense of it. One-way implications are also sufficient.