Genus fields of abelian extensions of congruence rational function fields

TL;DR

This paper constructs genus fields for congruence function fields, correcting previous proofs and providing explicit descriptions for various types of extensions, including Kummer, Artin--Schreier, and cyclic p-extensions.

Contribution

It offers a corrected and extended framework for genus fields in congruence function fields, including explicit constructions for several extension types.

Findings

Corrected proofs of key theorems in genus field theory.

Explicit descriptions of genus fields for Kummer, Artin--Schreier, and cyclic p-extensions.

Application of cyclotomic function field ideas to general cases.

Abstract

In the published version of this paper [Finite Fields and Their Applications {\bf 20} (2013) 40--54], there is an error in the proof of Theorem 4.2 of the paper. Here we correct the error and give the right statments for Theorems 4.2, 4.5 and 5.2 We give a construction of genus fields for congruence function fields. First we consider the cyclotomic function field case following the ideas of Leopoldt and then the general case. As applications we give explicitly the genus fields of Kummer, Artin--Schreier and cyclic $p$--extensions. Kummer extensions were obtained previously by G. Peng and Artin--Schreier extensions were obtained by S. Hu and Y. Li.