What are the p-adic algebraic numbers?

Question

"Given $p$, what are the elements of $\mathbb{Q}_p$ algebraic over $\mathbb{Q}$?"

I periodically wonder this and come across this mathoverflow question which seems to be asking the same thing. The chosen answer doesn't seem to answer that question (that I can see), and googling "p-adic algebraic numbers" returns that question as the top result. At that point I give up and wait until I forget and try again. So this time I'll ask:

Do you know of a (more convenient) characterization of $\overline{\mathbb{Q}}\cap\mathbb{Q}_p$ or have references for the "$p$-adic algebraic numbers?"

I'm not sure there's a characterization of "real algebraic numbers" much more satisfying than "real algebraic numbers," but the p-adic absolute value is inherently more "algebraic" than the real absolute value, and there are differences as $p$ varies, so what are they?

Answer 1

Let $O_\overline{\Bbb{Q}}$ be the algebraic integers, take some maximal ideal $\mathfrak{P}\subset O_\overline{\Bbb{Q}}$ containing $p$, let $G=\{ \sigma\in Gal(\overline{\Bbb{Q}}/\Bbb{Q}), \sigma(\mathfrak{P})=\mathfrak{P}\}$, then $G\cong Gal(\overline{\Bbb{Q}}_p/\Bbb{Q}_p)$ and $\Bbb{Q}_p\cap \overline{\Bbb{Q}}$ is (isomorphic to) the subfield of $\overline{\Bbb{Q}}$ fixed by $G$.

Equivalently, let $S$ be the set of (infinite degree) algebraic extensions $K/\Bbb{Q}$ for which some maximal ideal $\mathfrak{p}\subset O_K$ is such that $O_K/\mathfrak{p}\cong \Bbb{Z}/p\Bbb{Z},p\not \in \mathfrak{p}^2$. Then $\Bbb{Z}_p$ is (isomorphic to) the completion of $O_K$ at $\mathfrak{p}$, and $\Bbb{Q}_p\cap \overline{\Bbb{Q}}$ is (isomorphic to) any maximal element of $S$.