Suppose $L/k$ is a Galois extension of number fields, $G$ is the Galois group, and for $\frak p$ a prime ideal of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : $K$] = $p$ is prime. Suppose $m$ is the highest integer so $\mu_{p^m}\subseteq L$, and $\mu_{p^n}\subseteq K$ with $n\geq m-1$ [$\mu_{p^m}$=the group of $p^m$-th roots of unity]. We know $N_{L/K}(\mu_{p^m})\subseteq\mu_{p^n}$. How do we determine its exact image?

While I only need the more specific case in the paragraph above, I'm pretty sure I read about the (more general) question in the title in some well-known book; and surprisingly, I can't find any references for this on the internet.

Suppose that $L/k$ is a Galois extension of number fields and that $G$ is the corresponding Galois group. Further, for $\frak p$ a prime ideal of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : $K$] = $p$ is prime. Suppose $m$ is the highest integer such that $\mu_{p^m}\subseteq L$, and $\mu_{p^n}\subseteq K$ with $n\geq m-1$ [$\mu_{p^m}$=the group of $p^m$-th roots of unity]. We know that $N_{L/K}(\mu_{p^m})\subseteq\mu_{p^n}$. How do we determine its exact image?

While I only need the more specific case in the paragraph above, I'm pretty sure I read about the (more general) question in the title in some well-known book; and surprisingly, I can't find any references for this on the internet.