Max-spectrum of multivariate polynomial ring over an algebraically closed field

Let $k$ be an algebraically closed field. The max-spectrum of $k[x_{1},x_{2},\ldots ,x_{n}]$ is described below.

As a set

Further information: weak Nullstellensatz for algebraically closed fields

As a set, the max-spectrum is identified with $k^{n}$, where the identification is as follows:

$(a_{1},a_{2},\ldots ,a_{n})\mapsto (x_{1}-a_{1},\ldots ,x_{n}-a_{n})$

The left side denotes a point in $k^{n}$ and the right side denotes the maximal ideal for that point. Equivalently, we can think of the maximal ideal as the kernel of the "evaluation map" which sends a polynomial $p$ to the value $p(a_{1},a_{2},\ldots ,a_{n})$.

The above is one of the many formulations of the weak Nullstellensatz for algebraically closed fields.

As a topological space

The topology that we give to the max-spectrum coincides with the usual Zariski topology we put on $k^{n}$.