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

Question

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?

Thanks in advance.