The order of the reductions of an algebraic integer

2011/07/22 by Antonella Perucca, Perucca, Antonella
Mathematics · #11R18 #11R44 (primary) #11Y40 (secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1107.4595

openalex publication_date 2011/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or, more generally, has some prescribed l-adic valuation). We evaluate the degree over K of extensions of the form K(ζ_m, √[n]a) with n≤ m, which are obtained by adjoining a root of unity of order l^m and the lⁿ-th roots of a, as this is needed for computing the above density.

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