Work

Genus theory, governing field, ramification and Frobenius

Year: 2025

Type: article

Abstract: We develop, through a governing field, genus theory for a number field K with tame ramification in T and splitting in S, where T and S are finite disjoint sets of primes of K. This approach extends the one initiated by the second author in the case of the class group. We are able to express the S-T genus number of a cyclic extension L/K of degree p in terms of the rank of a matrix constructed from the Frobenius elements of the primes ramified in L/K, in the Galois group of the underlying governing extension. For quadratic extensions L/Q, the matrices in question are constructed from the Legendre symbols of the primes ramified in L/Q and the primes of S.

Source: Acta Arithmetica

DOI: 10.4064/aa240710-25-6

Authors: Roslan Ibara Ngiza Mfumu, Christian Maire

Institutions: Centre National de la Recherche Scientifique, Franche-Comté Électronique Mécanique Thermique et Optique - Sciences et Technologies, Université Marie et Louis Pasteur, Marien Ngouabi University

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funders: International Mathematical Union, Agence Nationale de la Recherche, Centre National de la Recherche Scientifique, Centers for Disease Control and Prevention

Awards: G2659019386, Ingénierie et Innovation par les sciences physiques, les savoir-faire technologiques et l'interdisciplinarité, G6803667113