Exercise 39, § 7

Let K be a field, L a subfield of K and $(K_\alpha)_{\alpha \in I}$ a left directed family of subfields of K such that $L = \bigcap_{\alpha \in I} K_\alpha$. Let E be a vector space over K and $(a_i)_{1 \leq i \leq m}$ a finite family of elements of E; show that if the family $(a_i)$ is free over L, there exists an index $\alpha$ such that $(a_i)$ is free over $K_\alpha$. (Argue by induction on $m - r$, where $r$ is the rank of the family $(a_i)$ over K.)