Proposition 3

— Let $\mathscr{F}$ be a set of subfields of a field E, directed with respect to the relation $\subset$. The union L of the fields of $\mathscr{F}$ is a field.

For if x and y are two elements of L, there exist two fields R, S of $\mathscr{F}$ such that $x \in R,\ y \in S$; let T be a field of $\mathscr{F}$ containing R and S; then $x \in T,\ y \in T$, hence $x + y,\ xy$ and $x^{-1}$ (if $x \neq 0$) belong to T, hence to L.