What are $\bar{\mathbb{Q}}$ and $\bar{\mathbb{Q}_{\ell}}$?

Question

Let $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$ and $\bar{\mathbb{Q}_{\ell}}$ the algebraic closure of $\mathbb{Q}_{\ell}$ ($\ell$ is an integer). Could we describe the elements in $\bar{\mathbb{Q}}$ and $\bar{\mathbb{Q}_{\ell}}$ explicitly? Here $\mathbb{Q}_{\ell}$ is the $p$-adic field at $\ell$. Thank you very much.

Answer 1

The fields $\overline{\mathbb{Q}_{\ell}}$ and $\mathbb{C}$ are indeed isomorphic as abstract fields, i.e., both are algebraically closed fields of characteristic zero and with transcendence degree $2^{ℵ_0}$ over $\mathbb{Q}$. Of course, this isomorphism does not preserve topology: $\overline{\mathbb{Q}_{\ell}}$ is not complete, for example, but $\mathbb{C}$ is. The completion of of $\overline{\mathbb{Q}_{\ell}}$ is $\mathbb{C}_{\ell}$. There are many desriptions about the algebraic closures, but it depends on what you want to know. Perhaps you find also the following question interesting in this context: How far are the $p$-adic numbers from being algebraically closed?.