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

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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.