This question arose while I was reading a paper I found in the web.
It might be very simple, but I don't know the answer.
Let $\mathbb{R}$ be the set of real numbers and $\mathbb{Q}_p$ the set of all $p$-adic numbers.
My question is: how can I construct (or at least guarantee the existence of) an algebraically closed field $\Omega$ of characteristic $0$ containing both $\mathbb{R}$ and $\mathbb{Q}_p$ for all primes $p$?
More generally, given a finite or infinite family of fields with the same characteristic (and possibly a common subfield), can I prove the existence of such a field? If not, under which conditions does it hold?
Thank you in advance for your help.
Edit: about my background, my level is basic; that is, I know what an algebraically closed field is and basic facts about Field Theory from a basic Galois theory course
Best Answer
It is possible to embed the algebraic closure of $\Bbb Q_p$ into $\Bbb C$, if you want. We also can consider the completion $\Bbb C_p$ of $\overline{\Bbb Q_p}$. This field is called the field of $p$-adic complex numbers.
The details have been already discussed at this site, e.g., here:
Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?
The embedding is guaranteed by the axiom of choice.