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Topological aspects of infinitude of primes in arithmetic progressions

Volume 140 / 2015

František Marko, Štefan Porubský 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.

Authors

  • František MarkoPennsylvania State University
    76 University Drive
    Hazleton, PA 18202, U.S.A.
    e-mail
  • Štefan PorubskýInstitute of Computer Science
    Academy of Sciences of the Czech Republic
    Pod Vodárenskou věží 2
    182 07 Praha 8, Czech Republic
    e-mail

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