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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 thepaper. 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. Firstwe consider the cyclotomic function field case following the ideas of Leopoldtand then the general case. As applications we give explicitly the genus fieldsof Kummer, Artin--Schreier and cyclic $p$--extensions. Kummer extensions wereobtained previously by G. Peng and Artin--Schreier extensions were obtained byS. Hu and Y. Li.

Genus fields of abelian extensions of congruence rational function fields