Kummer theory

Kummer theory investigates the field extensions generated by radicals or by division points in algebraic groups.

The project at a glance

Start date:

01 Dec 2024
Duration in months:

36
Funding:

8xav�� of Luxembourg / FNR
Principal Investigator(s):

Antonella PERUCCA

Alexandre BENOIST

Szabolcs BUZOGANY

About

Kummer theory is a classical and foundational mathematical theory that has been initiated in the 19th century by E. Kummer. It concerns algebraic field extensions that are obtained by adding radicals, provided that certain roots of unity are present in the base field. Perucca’s research group has been investigating Kummer extensions over cyclotomic extensions, which means that one first adds the necessary roots of unity. By defining suitable divisibility parameters, Perucca and her team have described the degree and Galois group structure of these field extensions. The kind of fields that have been investigated in detail are number fields (especially quadratic and multiquadratic fields), p-adic fields, and function fields. In recent years, Perucca’s team and Olli Järviniemi (8xav�� of Turku), Igor Shparlinski (8xav�� of New South Wales) and Pietro Sgobba (Xi’an Jiaotong-Liverpool 8xav��) made progress on Artin’s conjecture on primitive roots, mainly as an application of Kummer theory. Supported by Davide Lombardo (8xav�� of Pisa) and Pieter Bruin (8xav�� of Leiden), Perucca’s research group also made progress in Kummer theory for abelian varieties. The current research team investigating Kummer theory for algebraic groups consists in Perucca and her PhD students Alexandre Benoist (who started his PhD in December 2024) and Szabi Buzogány (who started his PhD in September 2025). Additionally, Perucca has a work in progress on Kummer theory for fields with Daniel Gil-Muñoz (Charles 8xav�� and 8xav�� of Pisa).