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