Reference : Kummer theory for number fields and the reductions of algebraic numbers II
E-prints/Working papers : First made available on ORBilu
Physical, chemical, mathematical & earth Sciences : Mathematics
http://hdl.handle.net/10993/36399
Kummer theory for number fields and the reductions of algebraic numbers II
English
Perucca, Antonella mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Sgobba, Pietro mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
2020
Yes
[en] number field ; reduction ; multiplicative order ; arithmetic progression ; density
[en] 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 \ell^e is a prime power and a is a multiple of \ell (and a is a multiple of 4 if \ell=2), then the density of primes p of K such that the order of (G mod p) is congruent to a modulo \ell^e only depends on a through its \ell-adic valuation.
Researchers
http://hdl.handle.net/10993/36399

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