The distribution of the multiplicative index of algebraic numbers over residue classes
- Resource Type
- Original Paper
- Authors
- Moree, Pieter; Perucca, Antonella; Sgobba, Pietro
- Source
- Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 94(1):1-17
- Subject
- Reductions of algebraic numbers
Multiplicative index and order
Primes in arithmetic progression
Natural density
Primary: 11R45
Secondary: 11A07, 11R44
- Language
- English
- ISSN
- 0025-5858
1865-8784
Let K be a number field and G a finitely generated torsion-free subgroup of K×pindp(G)(Gmodp)p. Given a prime K×pindp(G)(Gmodp)p of K we denote by K×pindp(G)(Gmodp)p the index of the subgroup K×pindp(G)(Gmodp)p of the multiplicative group of the residue field at K×pindp(G)(Gmodp)p. Under the Generalized Riemann Hypothesis we determine the natural density of primes of K for which this index is in a prescribed set S and has prescribed Frobenius in a finite Galois extension F of K. We study in detail the natural density in case S is an arithmetic progression, in particular its positivity.