Say we have a number field $K$ in which the rational prime $2$ splits like this $$(2)\mathcal{O}_K:=\mathfrak{P}_1\mathfrak{P}_2 .$$
I am wondering what is $K_{\mathfrak{P}_1}\cap K_{\mathfrak{P}_2}$. Where $K_{\mathfrak{P}_i}$ is the completion of $K$ at the prime $\mathfrak{P}_i$. Any help would be much appreciated!
The question makes no sense as written, because those completions are not initially inside a common larger field. And putting them into one might make the answer depend on the choice of embeddings into the larger common field.
It's like taking two finite extensions of a field $F$ and asking for their intersection. Unless the fields are already presented inside a common larger field, there is no intersection.
Suppose I say $\alpha^4 = 2$ and $\beta^4 = 2$ in fields of characteristic $0$, and I want to know $\mathbf Q(\alpha) \cap \mathbf Q(\beta)$. The question has no answer because I haven't given you a larger field containing these so that you can decide what lies in both of them. Inside $\mathbf C$, if $\beta = \pm \alpha$ then the fields are equal and their intersection is the full fields, while if $\beta \not= \pm \alpha$ then the intersection is $\mathbf Q(\sqrt{2})$. The exact same kind of answer could be given if $\alpha$ and $\beta$ are in an algebraic closure of $\mathbf Q_2$.