The elements in the composite field $FK$

Solution 1:

To answer your question on what $FK$ looks like, under the assumption that they both sit inside an (unspecified) immense field $\Omega$, let me point out that $FK$ may be thought of in three ways: $F(K)$, $K(F)$, and the intersection of all subfields of $\Omega$ that contain $F$ and $K$.

If $S$ is any subset of $\Omega$, the set $F(S)$ can be defined as the set of all $F$-rational expressions in finitely many elements of $S$. A moment’s thought convinces you that $F(S)$ is indeed a field, and a few more moments will convince you that this field is also the intersection of all subfields of $\Omega$ containing both $F$ and $S$. This shows you that the “three ways” I mention at the top do indeed describe the same field, and further describe what elements of $FK$ look like.