Kummer theory for number fields via entanglement groups

2021 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba ◽  
Sebastiano Tronto
Keyword(s):  
Number Fields ◽  
Kummer Theory

scholarly journals Kummer theory for number fields and the reductions of algebraic numbers

2019 ◽  
Vol 15 (08) ◽  
pp. 1617-1633 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba
Keyword(s):  
Number Field ◽  
Arithmetic Progression ◽  
Direct Proof ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Kummer Theory ◽  
Multiplicative Order ◽  
Higher Rank ◽  
Multiplicative Subgroup ◽  
Torsion Free

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if [Formula: see text] is a finitely generated and torsion-free multiplicative subgroup of a number field [Formula: see text] having rank [Formula: see text], then the ratio between [Formula: see text] and the Kummer degree [Formula: see text] is bounded independently of [Formula: see text]. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).

scholarly journals A note on Kummer theory of division points over singular Drinfeld modules

2001 ◽  
Vol 64 (1) ◽  
pp. 15-20 ◽  
Author(s):  
Anly Li
Keyword(s):  
Number Fields ◽  
Drinfeld Modules ◽  
Kummer Theory ◽  
Complete Analogy

In this paper, we shall establish a Kummer theory of division points over singular Drinfeld modules which is in complete analogy with the classical one in number fields.

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

2020 ◽  
Vol 15 (1) ◽  
pp. 75-92 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba
Keyword(s):  
Rational Number ◽  
Prime Power ◽  
Number Fields ◽  
Free Subgroup ◽  
Algebraic Numbers ◽  
Kummer Theory ◽  
Adic Valuation ◽  
Torsion Free Subgroup ◽  
Almost All ◽  
Torsion Free

AbstractLet 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.

scholarly journals Cutting towers of number fields

2021 ◽  
Author(s):  
Farshid Hajir ◽  
Christian Maire ◽  
Ravi Ramakrishna
Keyword(s):  
Number Fields

scholarly journals Theta series and number fields: theorems and experiments

2021 ◽  
Author(s):  
Adrian Barquero-Sanchez ◽  
Guillermo Mantilla-Soler ◽  
Nathan C. Ryan
Keyword(s):  
Theta Series ◽  
Number Fields

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

scholarly journals The 2-rank of the class group of some real cyclic quartic number fields

2021 ◽  
Vol 131 (1) ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohammed Tamimi ◽  
Abdelkader Zekhnini
Keyword(s):  
Class Group ◽  
Number Fields
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