Exercise 11, § 1

Assume that $k$ is algebraically closed. Let $\mathfrak{g}$ be a Lie algebra, V a simple $\mathfrak{g}$-module, and D the field of endomorphisms of the $\mathfrak{g}$-module V. Show that, if $k$ is uncountable,$^7$ then $D =k$. (If not, D would contain a subfield isomorphic to $k$(X), X being an indeterminate, and we would have dim$_kD>ℵ_0$, hence also dim$_kV>ℵ_0$, which is absurd since V is a monogenic $U(\mathfrak{g}$)-module.)