For a finite cyclic p-extension L/K of a rational function field K = k(x) over an algebraically closed field k of characteristic p > 0 such that every ramified prime divisor is fully ramified, we find a basis of the fc[G]-module CIL(0) of holomorphic differentials of L. We use this basis, which is similar to the Boseck-Garcia basis in the elementary abelian case, to find the fc[G]-module structure of ^£,(0) in terms of indecomposable modules.

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

Let k be a finite field. For a function field K over k and m > 3, it is proven that there are infinitely many non-isomorphic function fields L such that L/K is a separable extension of degree m and Aut£ L -{Id}. It is also shown that for a finite group G, there are infinitely many non-isomorphic function fields L/k such that Aut* L = G. Finally, given any finite nilpotent group G such that |G| > 1 and (|<7|, \k\ -1) = 1 and any function field K over k 9 there are infinitely many non-isomorphic function fields L over k with Gal(L/K) = Aut* L £* G.

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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.