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.



How locally ubiquitous are totally real fields?
The answer to the first question is "yes". See [this paper of the Dokchitser brothers](http://arxiv.org/abs/0906.1815), 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.


