<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>K</italic> be a number field, and let <italic>G</italic> be a finitely generated and torsion-free subgroup of <italic>K</italic><italic>×</italic>. For almost all primes p of <italic>K</italic>, we consider the order of the cyclic group (<italic>G</italic> mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓ<italic>e</italic> is a prime power and <italic>a</italic> is a multiple of ℓ (and <italic>a</italic> is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of <italic>K</italic> such that the order of (<italic>G</italic> mod 𝔭) is congruent to <italic>a</italic> modulo ℓ<italic>e</italic> only depends on <italic>a</italic> through its ℓ-adic valuation.</abstract>

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.

Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

Uniform distribution theory

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{"article-title":"Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II"}

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic...

Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

Publié en ligne: 24 juil. 2020

Pages: 75 - 92

Reçu: 07 juil. 2019

Accepté: 23 mars 2020

DOI: https://doi.org/10.2478/udt-2020-0004

Mots clés
Number field, reduction, multiplicative order, arithmetic progression, density

© 2020 Antonella Perucca et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

Publié en ligne: 24 juil. 2020

Pages: 75 - 92

Reçu: 07 juil. 2019

Accepté: 23 mars 2020

DOI: https://doi.org/10.2478/udt-2020-0004

Mots clésNumber field, reduction, multiplicative order, arithmetic progression, density

© 2020 Antonella Perucca et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Mots clés
Number field, reduction, multiplicative order, arithmetic progression, density