Is there an explicit description of the maximal tamely ramified extension of $\mathbf Q_p$?
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1$\begingroup$ You can find it in Chapter 16 of Hasse's Number Theory, not just for ${\bf Q}_p$ but also for any complete discretely valued field with finite residue field. $\endgroup$– Chandan Singh DalawatCommented Sep 19, 2012 at 2:53
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1 Answer
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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, in particular Theorem 11 and Corollary 12.)
answered Sep 18, 2012 at 16:12
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$\begingroup$ David, the Galois group of the maximal unramified extension of ${\bf Q}_p$ is $\hat{\bf Z}$, not $\hat{\bf Z}^\times$. $\endgroup$ Commented Sep 19, 2012 at 2:51
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$\begingroup$ Oops, that was silly! I have edited the answer to correct this, thanks for spotting it. $\endgroup$ Commented Sep 20, 2012 at 7:03