Genus fields of congruence function fields (Q730373)

Genus fields for congruence function fields were defined in 1987 by \textit{M. Rosen} [Expo. Math. 5, 365--378 (1987; Zbl 0632.12017)], and explicit constructions of them were provided for Kummer extensions by \textit{G. Peng} [J. Number Theory 98, No. 2, 221--227 (2003; Zbl 1049.11116)], and for Artin-Schreier extensions by \textit{S. Hu} and \textit{Y. Li} [Finite Fields Appl. 16, No. 4, 255--264 (2010; Zbl 1203.11076)]. Genus fields for abelian extensions of the rationals were described by \textit{H.-W. Leopoldt} [Math. Nachr. 9, 351--362 (1953; Zbl 0053.35502)] using Dirichlet characters. The authors describe in a similar way the genus fields of abelian extensions of rational congruence function fields using Dirichlet characters for cyclotomic function fields.