Equidistribution of realizable Steinitz classes for cyclic Kummer extensions

2025/06/15 by Brody Lynch, Lynch, Brody
Mathematics · #Advanced Topics in Algebra #Algebraic number #Algebraic number field #Algebraic structures and combinatorial models #Class (philosophy) #Class field theory #Discriminant #Equidistributed sequence #FOS: Mathematics #Field (mathematics) #Ideal class group #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2506.12999

openalex publication_date 2025/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

Let ℓ be prime, and K be a number field containing the ℓ-th roots of unity. We use classical algebraic number theory and some analytic techniques to prove that the Steinitz classes of \mathbb Z/ℓ\mathbb Z extensions of K ordered by relative discriminant are equidistributed among realizable classes in the ideal class group of K. For ℓ = 2, this was proved by Kable and Wright using the deep theory of prehomogeneous vector spaces. Foster proved that Steinitz classes are uniformly distributed between realizable classes for tamely ramified elementary-m extensions using the theory of Galois modules; our approach eliminates this tameness hypothesis.

Citations

• Counting biquadratic number fields with quaternionic and dihedral extensions
• Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces
• On counting rings of integers as Galois modules
• The density of discriminants of quintic rings and fields

Related

• The Steinitz Realization Problem
• Conjectures for distributions of class groups of extensions of number fields containing roots of unity
• Primes of Higher Degree
• Equidistribution of Elements of Norm 1 in Cyclic Extensions
• Generalized Steinberg Relations