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.