Let $p$ be a fixed prime number.

Question 1: Given a finite extension $K$ of $\mathbb{Q}_p$ is there a totally real extension $F$ of $\mathbb{Q}$ and a place $v$ of $F$ over $p$ such that $F_v = K$?

This is used in the proof of the local Langlands conjecture (thus I am quite sure that the answer is Yes) but I have never seen a reference. My state of knowledge is similar for the next question (and again I would be very grateful for a reference):

Question 2: Given an integer $g \geq 1$ is there a totally real extension $F$ of $\mathbb{Q}$ such that $F \otimes_{\mathbb{Q}} \mathbb{Q}_p = \mathbb{Q}_p \times \dots \times \mathbb{Q}_p$ ($g$ copies).

A common naive generalization of both questions is the following question (which is now a real question):

Question 3: Let $K_1, \dots, K_g$ be finite extensions of $\mathbb{Q}_p$. Is there a totally real extension $F$ of $\mathbb{Q}$ such that $F \otimes_{\mathbb{Q}} \mathbb{Q}_p = K_1 \times \dots \times K_g$?

I would not be surprised if the answer is No due to trivial reasons which I am just not seeing.

  • 6
    $\begingroup$ Have you tried using Krasner's Lemma, an obvious weak analogue for the reals, and the fact that $\mathbf{Q}$ is dense in $\mathbf{Q}_p\times\mathbf{R}$, to answer all of these questions affirmatively yourself? Seems to me like it shouldn't be so hard... $\endgroup$ – Kevin Buzzard Apr 27 '11 at 6:33
  • $\begingroup$ Ah, yes, I feel a little bit ashamed. That question was probably fired too quickly. Anyway, thanks for your and Alex' answer! $\endgroup$ – Torsten Wedhorn Apr 27 '11 at 9:37

The answer to the first question is "yes". See this paper of the Dokchitser brothers, Lemma 3.1 for the case where $K/\mathbb{Q}_p$ is Galois. In the general case, apply the result to the Galois closure $K'$ of $K$ to get $F'$, identify the Galois group of the local fields with a decomposition group $D$ at $p$ inside the global Galois group and take the fixed subfield of the subgroup of $D$ corresponding to $K$.

As Kevin says, unless I am missing something, the Dokchitsers' proof works with minor modifications for all three of your questions. Note that the result for question 3 follows from the previous two (using the slightly more general version of qn 1 in the above link): first take an extension in which $p$ is totally split, then work with each of the places above $p$ separately, using the answer to qn 1.

  • $\begingroup$ There is a more precise lemma in Clozel-Harris-Taylor, still with a short and simple proof; see lemma 4.1.2. It constructs a solvable Galois extension with prescribed completions at finitely many places (incl. the real place) that is linearly disjoint from any given finite Galois extension. I think there's another precise version which lets you prescribe the global degree (subject to constraints arising from Grünwald-Wang) in Artin-Tate, section on G-W. [I don't have the book with me to give you a precise reference.] $\endgroup$ – fherzig Apr 27 '11 at 12:24
  • $\begingroup$ @Florian Thank you for the extra references. In Artin-Tate, I have found Theorem 5 on page 105, which deals with cyclic extensions. $\endgroup$ – Alex B. Apr 27 '11 at 13:06
  • $\begingroup$ @Alex: Thanks for checking, that makes sense that they deal with cyclic extensions only. $\endgroup$ – fherzig Apr 27 '11 at 18:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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