Exercise 2, § 3

Let $(E_i)_{i \in I}$, be a family of extensions of a field K, contained in an extension L of K. If $F_i$ is the relative algebraic closure of K in $E_i$, show that the relative algebraic closure of K in $E = \bigcap_{i \in I} E_i$ is $F = \bigcap_{i \in I} F_i$.