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Congruences of Ankeny-Artin-Chowla type and the p-adic class number formula revisited

100%

Marko F.

Acta Arithmetica

2015

tom 167

nr 3

281-298

The purpose of this paper is to interpret the results of Jakubec and his collaborators on congruences of Ankeny-Artin-Chowla type for cyclic totally real fields as an elementary algebraic version of the p-adic class number formula modulo powers of p. We show how to generalize the previous results to congruences modulo arbitrary powers $p^t$ and to equalities in the p-adic completion $ℚ_p$ of the field of rational numbers ℚ. Additional connections to the Gross-Koblitz formula and explicit congruences for quadratic and cubic fields are given.

Topological aspects of infinitude of primes in arithmetic progressions

63%

Marko F., Porubský Š.

Colloquium Mathematicum

2015

tom 140

nr 2

221-237

We investigate properties of coset topologies on commutative domains with an identity, in particular, the 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 𝓢-coprime topologies on ℤ. Finally, we give a new proof for the infinitude of prime ideals in number fields.

Ograniczanie wyników

1 Acta Arithmetica

1 Colloquium Mathematicum

2 Marko F.

1 Porubský Š.

2 2015

Wyniki wyszukiwania

Congruences of Ankeny-Artin-Chowla type and the p-adic class number formula revisited

Topological aspects of infinitude of primes in arithmetic progressions