Topological aspects of infinitude of primes in arithmetic progressions
Volume 140 / 2015
Colloquium Mathematicum 140 (2015), 221-237
MSC: Primary 11B05; Secondary 11N80, 11N25, 11A25, 22A99, 22A15.
DOI: 10.4064/cm140-2-5
Abstract
We investigate properties of coset topologies on commutative domains with an identity, in particular, the $\mathcal {S}$-coprime topologies defined by Marko and Porubský (2012) and akin to the topology defined by Furstenberg (1955) in his proof of the infinitude of rational primes. We extend results about the infinitude of prime or maximal ideals related to the Dirichlet theorem on the infinitude of primes from Knopfmacher and Porubský (1997), and correct some results from that paper. Then we determine cluster points for the set of primes and sets of primes appearing in arithmetic progressions in $\mathcal {S}$-coprime topologies on $\mathbb {Z}$. Finally, we give a new proof for the infinitude of prime ideals in number fields.
