A complex affine algebraic set is either finite or has the cardinality of $\mathbb C$. One proof takes a positive-dimensional irreducible component and applies <Noether normalization>: the resulting finite dominant map to a positive-dimensional affine space has a point above every closed point.


Complex affine algebraic set cardinality dichotomy

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Every finitely generated algebra $A$ over a field has algebraically independent elements $y_1,\ldots,y_d$ such that $A$ is finite as a module over the polynomial subalgebra $k[y_1,\ldots,y_d]$. The integer $d$ is the Krull dimension of $A$.


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 Incoming links: Noether normalization