Existence of noncanonical large field Let $K, L, E$ be fields with embeddings $K \rightarrow L, K \rightarrow E$.  There is in general no canonical field $\Omega$ which contains both $L$ and $E$ as $K$-subfields.  Under certain conditions the tensor product $L \otimes_K E$ will be a field, and one can use this.  
However, does there always exist some field $\Omega$ which contains $L$ and $E$ as $K$-subfields?

Yes, there does.  Note that the ring $L\otimes_K E$ is nonzero, since it is the tensor product of two nonzero vector spaces over a field.  So it has a maximal ideal $M$.  The quotient $\Omega=L\otimes_K E/M$ is then a field, and the canonical inclusions $L\to L\otimes_K E$ and $E\to L\otimes_K E$ give embeddings of $K$-algebras $L\to \Omega$ and $E\to \Omega$.
Alternatively, take $\Omega\supseteq K$ to be algebraically closed and have large transcendence degree over $K$ (at least as large as the transcendence degrees of $L$ and $E$).  Picking a transcendence basis $B$ for $L$ over $K$, you can embed $K(B)$ into $\Omega$, and this extends to $L$ since $L$ is algebraic over $K(B)$ and $\Omega$ is algebraically closed.  By the same argument there is also an embedding $E\to\Omega$ over $K$.
