In wikipedia Zariski's lemma is the following
Let $K$ be a finitely generated $k$-algebra and suppose that $K$ is a field. Then $K$ is a field extension of $k$.
But if $M$ be a maximal ideal and $A$ is a finitely generated $C$-algebra($C$= complex numbers) then one can consider the induced surjective homomorphism from $A$ to $A/M$ and therefore $A/M$ is also a finitely generated $C$-algebra but $A/M$ is a field so $A/M$ should be equal $C$ which is not since its elements are cosets. So I am wondering does Zariski lemma mean that $K$ is isomorphic to a finite field extension of k?