MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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.)

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.