What is $\mathbb{Q}\_p \cap \overline{\mathbb{Q}}$ ? 


For instance, we know that $\mathbb{Q}\_p$ contains the $p-1$st roots of unity, so we might say that $\mathbb{Q}(\zeta) \subset \mathbb{Q}\_p \cap \overline{\mathbb{Q}}$, where $\zeta$ is a primitive $p-1$st root. 


As a more specific example, $x^2 - 6$ has 2 solutions in $\mathbb{Q}\_5$, so we could also say that $\mathbb{Q}(\sqrt{6},\sqrt{-1})\subset \mathbb{Q}\_p \cap \overline{\mathbb{Q}}$. 


**Edit:** I removed the motivation for this question (which I think stands by itself), as it will be better as a separate question once I think it through a bit better. 



Which p-adic numbers are also algebraic?
The field $K\_p = \mathbb{Q}\_p \cap \overline{\mathbb{Q}}$ is a very natural and well-studied one. I can throw some terminology at you, but I'm not sure exactly what you want to know about it.


1) It is often called the field of "$p$-adic algebraic numbers". This comes up in model theory: it is a $p$-adically closed field, which is the $p$-adic analogue of a real-closed field. In particular, it is elementarily equivalent to $\mathbb{Q}\_p$.


2) It is the Henselization of $\mathbb{Q}$ with respect to the $p$-adic valuation, or the 
fraction field of the Henselization of the ring $\mathbb{Z}\_{(p)}$ -- i.e., $\mathbb{Z}$ localized at the prime ideal $p$. 


The idea is that this field is not complete but is Henselian -- it satisfies the conclusion of Hensel's Lemma. Alternately and somewhat more gracefully, Henselian valued fields are characterized by the fact that the valuation extends uniquely to any algebraic field extension. 


Roughly speaking, Henselian fields are as good as complete fields for algebraic constructions but are not "big enough" to have the same topological properties. For instance, note that $K\_p$ cannot possibly be complete with respect to the $p$-adic valuation, because it is countably infinite and without isolated points: apply the Baire Category Theorem. 


