Teoria dei numeri Kummer Theory for Number Fields.
On Schinzel-W´ojcik problem. Pietro SGOBBA
Mohammed ANWAR
12-11-2018 - 15:00 Largo San Leonardo Murialdo,1 - Pal.C - Aula 311 15-16 Pietro Sgobba
Title: Kummer Theory for Number Fields.
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 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( √n G) : K(ζn)] is bounded independently of n. This result can be applied 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).
16-17 Mohamed Anwar
Title: On Schinzel-W´ojcik problem.
Abstract: The Schinzel–W´ojcik problem consists in determming if Given a1, . . . , ar ∈ Q∗ − {±1}, there exist infinitely many primes p such that they have the same multiplicative order modulo p. I introduce, under the assumption of Hypothesis H of Schinzel, a characterization of the r-tuples of rational numbers for which the Schinzel-W´ojcik problem has an affimative answer org: PAPPALARDI Francesco
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