Finitely generated $k$-algebra of Krull-dimension one has infinitely many primes

Question

I am looking for a reference or a quick proof of the statement that

Every finitely generated $k$-algebra of Krull-dimension one has infinitely many prime ideals.

Here $k$ is a field.

Answer 1

Expanding user26857's comment into an answer:

Let $R$ be a finitely generated $k$-algebra of dimension at least one, $k$ an arbitrary field. (The result does not require the dimension 1 hypothesis but does require that $R$ not be dimension zero.)

By Noether normalization, $R$ has a subring $S$ such that (a) $S$ is a polynomial ring (i.e. generated over $k$ by a finite number of algebraically independent elements), and (b) $R$ is finite over $S$. Because the dimension of $R$ is at least one, $S$ has at least one indeterminate.

(b) implies $R$ is integral over $S$, and the lying-over theorem then implies $R$ has at least one prime ideal for each prime ideal of $S$.

Thus the problem is reduced to proving the polynomial ring $S$ (with at least one indeterminate) has infinitely many prime ideals. Since any irreducible polynomial generates a prime ideal, and no two distinct monic polynomials can generate the same ideal, it will be sufficient to exhibit infinitely many irreducible monic polynomials.

At this point, Euclid's proof of the infinitude of the primes can be reappropriated to complete the proof. Given any finite list of monic irreducible polynomials $f_1,\dots,f_r$, the polynomial $f_1\dots f_r + 1$ is also monic, and it cannot have an irreducible factor already in the list $f_1,\dots,f_r$. Thus, any of its monic irreducible factors will extend the list. Thus there are infinitely many distinct monic irreducible polynomials in any polynomial ring, and therefore infinitely many prime ideals.