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 and to equalities in the p-adic completion of the field of rational numbers ℚ. Additional connections to the Gross-Koblitz formula and explicit congruences for...
Let be a totally real cyclic number field of degree that is the product of two distinct primes and such that the class number of the -th cyclotomic field equals 1. We derive certain necessary and sufficient conditions for the existence of a Minkowski unit for .
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...
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