Explicit Kummer theory for the rational numbers

Abstract

Let [Formula: see text] be a finitely generated multiplicative subgroup of [Formula: see text] having rank [Formula: see text]. The ratio between [Formula: see text] and the Kummer degree [Formula: see text], where [Formula: see text] divides [Formula: see text], is bounded independently of [Formula: see text] and [Formula: see text]. We prove that there exist integers [Formula: see text] such that the above ratio depends only on [Formula: see text], [Formula: see text], and [Formula: see text]. Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).

Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields

“…In [7,12] we have described a finite procedure for the computation of the --adelic failure over Q and over quadratic number fields. Now we consider number fields which are either multiquadratic or quartic cyclic, and we provide an explicit finite procedure to compute the -adelic failure B(M, n ) for all prime numbers , all n 1, and for all M 1 such that n divides M , see Sections 8 and 9.…”

Unified treatment of Artin-type problems

“…The group a has separated Kummer extensions w.r.t. Z: following [16,17] we have to consider the -adic failure [we have α…”

Division in modules and Kummer theory

“…If G = G m is the multiplicative group, extensions of this kind are studied by classical Kummer theory. Explicit results for this case can be found for example in [12], [14], [15] and [13]. The more general case of an extension of an abelian variety by a torus is treated in Ribet's foundational paper [16].…”