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.

topologicalcopies of $\mathbb{R}$ and $\mathbb{Q}_p$, since each of these induce distinct topologies on $\mathbb{Q}$. The isomorphism Ralph describes isnotcontinuous. – Kevin Ventullo Feb 7 '12 at 3:05