Geodesic
Topological aspects of infinitude of primes in arithmetic progressions
František Marko 1   ; Štefan Porubský 2
1 Pennsylvania State University 76 University Drive Hazleton, PA 18202, U.S.A.
2 Institute of Computer Science Academy of Sciences of the Czech Republic Pod Vodárenskou věží 2 182 07 Praha 8, Czech Republic
Colloquium Mathematicum, Tome 140 (2015) no. 2, pp. 221-237 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/cm140-2-5
Keywords: investigate properties coset topologies commutative domains identity particular mathcal coprime topologies defined nbsp marko nbsp porubsk nbsp akin topology defined furstenberg his proof infinitude rational primes extend results about infinitude prime maximal ideals related dirichlet theorem infinitude primes knopfmacher porubsk correct results paper determine cluster points set primes sets primes appearing arithmetic progressions mathcal coprime topologies mathbb finally proof infinitude prime ideals number fields

František Marko  1   ; Štefan Porubský  2

1 Pennsylvania State University 76 University Drive Hazleton, PA 18202, U.S.A.
2 Institute of Computer Science Academy of Sciences of the Czech Republic Pod Vodárenskou věží 2 182 07 Praha 8, Czech Republic 
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František Marko; Štefan Porubský. Topological aspects of infinitude of primes in arithmetic progressions. Colloquium Mathematicum, Tome 140 (2015) no. 2, pp. 221-237. doi: 10.4064/cm140-2-5

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