Definition of $\mathbb Q^c_p$

Question

Let $p$ be a prime number and let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q $ in $\mathbb C$, i.e. the field of algebraic numbers.

Is it possible at all to define the $p$-adic completion of $\mathbb Q^c$?

Answer 1

No. There is not a canonical p-adic completion of $\mathbb{Q}^c$, because there is not a canonical p-adic valuation to complete in.

The problem is that there are lots of (indeed uncountably many) possible valuations on $\mathbb{Q}^c$ extending the p-adic valuation on $\mathbb{Q}$. If you pick one of these (using the axiom of choice), call it $v$, and complete with respect to $v$, you get a completion $(\mathbb{Q}^c)_v$ of $\mathbb{Q}^c$. If $w$ is another choice of valuation extending the p-adic one, then the fields $(\mathbb{Q}^c)_v$ and $(\mathbb{Q}^c)_w$ are isomorphic, but not canonically so: there is no natural map between them.

One can validly say that p-adic completions (note plural!) of $\mathbb{Q}^c$ exist, but are only unique up to non-unique isomorphism.