Genus fields of abelian extensions of rational congruence function fields

Abstract

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

Genus fields of congruence function fields

“…We are interested in k * itself. In particular we have k * = k * ge and since k * /k is abelian, the description of such k * may be found in [14].…”

“…We also recall two results. With respect to the genus field of a finite abelian extension of k, we have the following results (see [14,15,Theorem 4.2]).…”

“…In this case s = t and N = P 1 · · · P t . Since for any prime P in k, the ramification index in K/k is the same as the ramification index in E/k (see [14,Section 4.1]) and F 0 = P∈P * k F P , we have E ge ⊆ F 0 . The infinite prime decomposes fully in F 0 ∩ R + /k.…”

GENUS FIELDS OF CYCLIC l-EXTENSIONS OF RATIONAL FUNCTION FIELDS

“…In [12] the genus field of a finite geometric abelian extension of k := F q (T ) was described and as applications the genus fields of cyclic extensions of prime degree over k were found explicitly. The results of Peng and of Hu and Li can be obtained in this way.…”

“…In this paper we use the results obtained in [12] to describe explicitly the genus field of cyclic extensions of degree l n where l n | q − 1. The case n = 1 is the result of Peng.…”