How locally ubiquitous are totally real fields?

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.

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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... –  Kevin Buzzard Apr 27 '11 at 6:33
Ah, yes, I feel a little bit ashamed. That question was probably fired too quickly. Anyway, thanks for your and Alex' answer! –  Torsten Wedhorn Apr 27 '11 at 9:37

1 Answer

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.

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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.] –  fherzig Apr 27 '11 at 12:24
@Florian Thank you for the extra references. In Artin-Tate, I have found Theorem 5 on page 105, which deals with cyclic extensions. –  Alex B. Apr 27 '11 at 13:06
@Alex: Thanks for checking, that makes sense that they deal with cyclic extensions only. –  fherzig Apr 27 '11 at 18:22