Let $L_\mathfrak q/K_\mathfrak p$ be an extension of non-Archimedean local fields. Here $L/K$ is an extension of number fields. $L_\mathfrak q/K_\mathfrak p$ is tamely ramified if and only if $L_\mathfrak q/L_\mathfrak q^1$ is totally tamely ramified. Here $L_\mathfrak q^1$ is the maximal unramified extension. Let $n=[L_\mathfrak q:L_\mathfrak q^1]$. $L_\mathfrak q/L_\mathfrak q^1$ is totally tamely ramified if and only if $L_\mathfrak q=L_\mathfrak q^1(\sqrt[n]\pi)$ with some uniformizer $\pi\in L_\mathfrak q^1$.
Neukirch's book defines the maximal tamely ramified extension to be the composite field of tamely ramified fields. However, is there only finitely many tamely ramified subextensions? If not, why the composite field is tamely ramified (since we checked for two fields case in Neukirch's book)?
