# Algebra of Algebraically Closed Field that is a field but not finitely generated as a module.

I was trying to understand a proof of the Weak Nullstellensatz. In the build-up to the proof it states the following Theorem, without proof.

If K is an algebraically closed field, then any K-algebra cannot be both a field and finitely generated as a K-module.

I'm not sure why this is true. Can somebody help me understand the proof to this?

• If $K\subset L$ is an algebraic field extension, and $K$ is algebraically closed, what can you say? – user26857 Apr 27 at 19:52