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Let $k$ be an algebraicly closed field.

and let $\mathbb A^n(k)$ be our affine n space:

For $n=1$ we can clasify our closed zariski subsets of 1-space:

These are finite point sets and whole space and empty set.

For $n\ge 2$ what is the good approach to understand general closed subsets? I am revising my commutative algebra course and a bit confused and want to see these closed subsets with intuitive and good explanatory way.

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    The closed subsets are of the form $V(I)$ for an ideal $I$ of $k[x_1,\ldots,x_n]$. Otherwise I don't know of a nice description that works for $n\geq 2$ in general. The $n=1$ case is very simple since, other than the zero ideal, the only prime ideals of $k[x]$ are maximal. – Dave Jan 26 '22 at 19:36
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    See https://math.stackexchange.com/questions/56916/what-do-prime-ideals-in-kx-y-look-like for $n=2$. – Kenta S Jan 26 '22 at 20:21

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