In this paper, we find the genus field of finite abelian extensions of the global rational function field. We introduce the term conductor of constants for these extensions and determine it in terms of other invariants. We study the particular case of finite abelian $p$-extensions and give an explicit description of their genus field.
"Genus fields of abelian extensions of rational congruence function fields, II." Rocky Mountain J. Math. 48 (7) 2099 - 2133, 2018. https://doi.org/10.1216/RMJ-2018-48-7-2099