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.



Does a "composite field" always exist?
The tensor product $F\_1 \otimes\_F F\_2$ is not 0, hence it has a quotient which is a field. This contains the images of both $F\_i$.


