Consider $k$-Zariski topology on $C^{n}$ ($C$ algebraically closed). Let $M$ be a maximal ideal of $C[x_{1},...,x_{n}]$ and $S$ the set of closed subsets of $C^{n}$ relative to $k$-Zariski topology.
Since $M$ is maximal ideal of $C[x_{1},...,x_{n}]$, follows that $Z(M)$ is a minimal non-empty subset of $S$?
$\to$ $Z(M)$ is a "zero" of $M$
This seems true for me and I want to use this as an argument in a proof, but I couldn't prove. Any hint?
Hint: use Zariski's lemma, which says that if $E$ is a finitely generated $C$-algebra which is also a field, then $E \cong C$ (in general a finite extension of $C$, but $C$ is algebraically closed). In particular, think about the map $C[x_1,\dots,x_n] \twoheadrightarrow C[x_1,\dots,x_n]/M$. Can you see how to proceed from here to explicitly describe the set $Z(M)$?