A very common construction in algebra is to consider the object "generated" by a subset or collection of subobjects: In group theory we look at the subgroup generated by a subset, in ring theory we look at the ideal generated by a subset, and in field theory we look at composite fields. Usually there end up being two equivalent ways to look at these objects: One describes the generated object as satisfying a minimality condition, and the other describes the elements of the generated object concretely as "algebraic combinations" of the elements of the generating set.
For example, if $R$ is a commutative ring and $A \subset R$ is a subset, then the ideal $(A) \subset R$ generated by $A$ can be described as:
(i) The smallest ideal of $R$ containing $A$, or
(ii) The collection of all finite $R$-linear combinations of elements of $A$, that is, $(A) = \{r_1a_1 + \cdots + r_na_n : r_1,\ldots,r_n \in R, a_1,\ldots, a_n \in A, n \in \mathbb{Z}^{\geq0}\}.$
We can do a similar thing in group theory by considering words obtained from the generating set, but I'm having trouble getting a description in the spirit of (ii) for a composite field. Of course, a description in the spirit of (i) is easy: If $F_1,F_2 \subset K$ are subfields, then the composite $F_1F_2$ is the smallest subfield of $K$ containing $F_1$ and $F_2$. It's harder, however, to represent the elements of $F_1F_2$ cleanly as "algebraic combinations" of the elements of $F_1F_2$ because division throws a wrench in the whole thing.
What's the cleanest way to put all the elements of $F_1F_2$ into nice closed expressions of elements of $F_1$ and $F_2$ as is done in the case of rings? I'm thinking quotients of linear combinations might do the trick, but I'm not sure.