Kummer theory for number fields and the reductions of algebraic numbers

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 (without relying on the Bashmakov–Ribet method) of the fact that if [Formula: see text] is a finitely generated and torsion-free multiplicative subgroup of a number field [Formula: see text] having rank [Formula: see text], then the ratio between [Formula: see text] and the Kummer degree [Formula: see text] is bounded independently of [Formula: see text]. We then apply this result to generalize 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).

The degree of Kummer extensions of number fields

“…This is a classical result of Kummer theory, which also holds more generally (under some assumptions) for products of abelian varieties and tori, see [2, Theorem 1] and [14] (see also [5,Lemme 14] and [1,Theorème 5.2]). Notice that Perucca and Sgobba gave an elementary proof for the existence of this constant, see [11,Theorem 3.1]. As a consequence, we have the following general result:…”

“…It is sufficient to apply (10) with our choice of s. For = 2 and ζ 4 / ∈ K we take s 2, so that the Kummer degrees appearing in (11) can be computed by Theorem 13. Notice that the equality (11) holds also in the case s < ω (where ω is as in Theorem 13) because if 1 m < ω we may replace m by ω.…”

“…For m and n fixed, this intersection is abelian over K and clearly contains ζ m . However, if is such that ζ / ∈ K, then by Schinzel's theorem on abelian radical extensions (see for instance [11,Theorem 3.3]), an element a ∈ K is such that the extension K(ζ n , n √ a) is abelian over K if and only if it is an n -th power in K. Therefore, in the remaining part of this section we consider a prime number such that ζ ∈ K.…”

“…Let s be as in (12). By [11,Theorem 2.7] there is a basis of G as a Z-module consisting of strongly -independent elements for all but finitely many primes , so that parameters for the -divisibility of G in K might be not all zero only for finitely many primes .…”

“…The finite set S of these primes can be computed by [11,Proof of Theorem 2.7]. Following this reference, the set S consists of the primes such that ζ ∈ K and those involved in the following computations.…”