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Throughout, we take as a complex subring (with unity). In this article, we’ll be interested in natural analogues of Euclid’s proof of the infinitude of the primes (i.e. the case . In particular, we’ll show that Euclid’s proof (and generalizations to this method) can be recast as a relationship between the size of the unit group of and , the number of prime ideals in. *(Here, denotes the spectrum of .)*

For the sake of explicit analogy, we include a (needlessly abstracted) version of Euclid’s result now:

**Theorem 1 (Euclid): ** *The integers have infinite spectrum.*

*Proof: *If not, let exhaust the list of prime ideals. As the integers form a PID, (in fact, they form a Euclidean domain), we may associate to each given ideal a prime generator such that . Let ; as is not a unit (replacing with if necessary), it admits a prime factor which equals some . But then divides both and , whence as well. This is a contradiction, and our result follows.

It is worth remarking that the “PID-ness” of the integers is not needed in the derivation of Theorem 1. Indeed, if were not principal, we may take instead to be *any* non-zero element of . Thus, we see that the PID-ness of is *not* the crucial property of the integers (viewed as a subfield of ) upon which our proof stands. Rather — as we’ll see after the fold — Euclid’s proof frames the infinitude of primes as a consequence of the *finiteness of the units group!*