**Part 1: a single finite place.** Let $K$ be a finite extension of $\mathbf{Q}\_p$.



> 
> Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $F$ at $v$ we have a topological isomorphism of topological field extensions of $\mathbf{Q}\_p$:
> 
> 
> $$F\_v\simeq K ?$$
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> 
> 


In imprecise words, does any $p$-adic field come as a completion of a number field?


**Part 2: a family of $p$-adic fields. (ASKED AS A SEPARATE QUESTION, [here](https://mathoverflow.net/questions/311196/interpolation-of-families-of-local-fields))** Let $(K\_p)\_p$ be a collection of $p$-adic fields. Assume the first part of the question has positive answer.
Does there exist a condition on $(K\_p)\_p$ such that there is one number field $K$ recovering each $K\_p$ as a completion?



“Algebraization" of $p$-adic fields
The answer to part 1 is yes. Given $K/\mathbb{Q}\_p$, let $\alpha\in K$ be a primitive element, with minimal monic polynomial $f(x)=x^n+\sum\_{i=1}^n a\_i x^{n-i}$, $a\_i\in\mathbb{Q}\_p$. So we have $K\cong\mathbb{Q}\_p[x]/(f(x))$. Krasner's lemma implies there exists a positive integer $N$ such that if $b\_i\in\mathbb{Q}\_p$ satisfy $v\_p(b\_i-a\_i)\geq N$, then the polynomial $g(x)=x^n+\sum\_i b\_i x^{n-i}$ satisfies $\mathbb{Q}\_p[x]/(f(x))\cong\mathbb{Q}\_p[x]/(g(x))$. In particular, since $\mathbb{Q}$ is dense in $\mathbb{Q}\_p$, we can choose the $b\_i$ to be rational. Then $F:=\mathbb{Q}[x]/(g(x))$ is a finite extension of $\mathbb{Q}$, and $F\otimes\mathbb{Q}\_p\cong\mathbb{Q}\_p[x]/(g(x))\cong K$. In general, tensoring a number field with $\mathbb{Q}\_p$ gives the product of the completions of that number field at the primes lying over $p$, so in this case there is only one prime $v$ of $F$ over $p$, and $F\_v\cong K$.


I do not know an answer to part 2, but I will mention some necessary conditions on the family $(K\_p)$: only finitely many can be ramified, only finitely many degrees occur, and $\{p:K\_p=\mathbb{Q}\_p\}$ has positive lower density.


