TITLE: Does a "composite field" always exist?
QUESTION [5 upvotes]: Suppose $F$ is a field, and $F_1, F_2$ are two extension fields of $F$.  Is it always the case that there is a field $L$, containing three subfields $F, K_1, K_2$ and two ring isomorphisms $\varphi_{i}:F_i\rightarrow K_1$ fixing $F$?  
Note 1: We lose no generality assuming $F$, rather than an isomorphic copy of $F$, is a subfield of $L$.
I ask this because I was wondering if there is a way to combine the reals and the $p$-adic numbers into a single extension of $\mathbb{Q}$.  
Note 2: I seem to recall someone telling me this couldn't be done (perhaps with additional topological data preserved).  But I cannot seem to remember the reason why.  In any case, I want to know if there is something other than topology which prevents it.

REPLY [7 votes]: In the language of Model Theory, your question can be rewritten as: "does the theory of fields have the amalgamation property? and the answer is yes.
Well known examples of theories with the amalgamation property include: fields, ordered fields, groups, abelian groups and boolean algebras.