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

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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. – Chandan Singh Dalawat Sep 19 '12 at 2:53
up vote 11 down vote accepted

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, in particular Theorem 11 and Corollary 12.)

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David, the Galois group of the maximal unramified extension of ${\bf Q}_p$ is $\hat{\bf Z}$, not $\hat{\bf Z}^\times$. – Chandan Singh Dalawat Sep 19 '12 at 2:51
Oops, that was silly! I have edited the answer to correct this, thanks for spotting it. – David Loeffler Sep 20 '12 at 7:03

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