Kummer theory for number fields and the reductions of algebraic numbers

Antonella Perucca, Pietro Sgobba

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)

Abstract

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 G is a finitely generated and torsion-free multiplicative subgroup of a number field K having rank r, then the ratio between nr and the Kummer degree [K(ζn,Gn): K(ζn)] is bounded independently of n. 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).

Original languageEnglish
Pages (from-to)1617-1633
Number of pages17
JournalInternational Journal of Number Theory
Volume15
Issue number8
DOIs
Publication statusPublished - 1 Sept 2019
Externally publishedYes

Keywords

  • Kummer theory
  • density
  • multiplicative order
  • number field
  • reduction

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