In the published version of this paper [Finite Fields and Their Applications 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 5.2We 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

ABSTRACT. Let k be a rational congruence function field and consider an arbitrary finite separable extension K/k. If for each prime in k ramified in K we have that at least one ramification index is not divided by the characteristic of K, we find the genus field Kge, except for constants, of the extension K/k. In general, we describe the genus field of a global function field.

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There is an error in the proof of Theorem 4.2 of the paper. Here we correct the error and give the right statements for Theorems 4.2, 4.5 and 5.2.