The Euclidean criterion for irreducibles

TL;DR

This paper generalizes Euclid's proof to establish a criterion for domains to have infinitely many atoms, connecting classical and topological proofs, and applies even in domains with non-unique factorizations.

Contribution

It introduces a Euclidean Criterion for irreducibles that extends Euclid's proof to broader algebraic domains, linking it with Furstenberg's topological approach.

Findings

The criterion applies to domains with non-unique factorizations.

Connections established between Euclidean and topological proofs.

Criterion identifies infinite irreducibles in various algebraic structures.

Abstract

We recast Euclid's proof of the infinitude of prime numbers as a Euclidean Criterion for a domain to have infinitely many atoms. We make connections with Furstenberg's "topological" proof of the infinitude of prime numbers and show that our criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles.