Article (Scientific journals)
Kummer theory for number fields and the reductions of algebraic numbers
Perucca, Antonella; Sgobba, Pietro
2019In International Journal of Number Theory
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Abstract :
[en] 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 n^r and the Kummer degree [K(\zeta_n,\sqrt[n]{G}):K(\zeta_n)] is bounded independently of n. We then apply this result to generalise 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).
Disciplines :
Mathematics
Author, co-author :
Perucca, Antonella  ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
Sgobba, Pietro ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
External co-authors :
no
Language :
English
Title :
Kummer theory for number fields and the reductions of algebraic numbers
Publication date :
2019
Journal title :
International Journal of Number Theory
ISSN :
1793-0421
Publisher :
World Scientific Publishing Co., Singapore
Peer reviewed :
Peer Reviewed verified by ORBi
Available on ORBilu :
since 18 June 2018

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