Is there an explicit description of the maximal tamely ramified extension of $\mathbf Q_p$?
Thank you.
Is there an explicit description of the maximal tamely ramified extension of $\mathbf Q_p$? Thank you. 


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://math.uga.edu/~pete/8410Chapter4.pdf, in particular Theorem 11 and Corollary 12.) 

