Is there an explicit description of the maximal tamely ramified extension of $\mathbf Q\_p$?



Maximal tamely ramified extension of $\mathbf Q_p$
Yes, there is. The maximal unramified extension is obtained by adding all roots of unity of order prime to $p$. The maximal tame extension is obtained by adding on top of that all $n$-th roots of $p$, for $n$ prime to $p$; so its Galois group is isomorphic to $\prod\_{\ell \ne p} \mathbf{Z}\_\ell$, and conjugation by $\operatorname{Gal}(\overline{\mathbf{Q}}\_p^{nr} / \mathbf{Q}\_p)\cong \widehat{\mathbf{Z}}$ acts on each $\mathbf{Z}\_\ell$ factor via the $\ell$-adic cyclotomic character.


(EDIT: For a reference for this see Pete Clark's notes at [http://alpha.math.uga.edu/~pete/8410Chapter4.pdf](http://alpha.math.uga.edu/%7Epete/8410Chapter4.pdf), in particular Theorem 11 and Corollary 12.)


