Genus fields of Kummer ℓn-cyclic extensions

Abstract

We give a construction of the genus field for Kummer ℓn-cyclic extensions of rational congruence function fields, where ℓis a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer ℓ-cyclic extension. Finally, we study the extension (K1K2)/(K1)(K2), for K1, K2 abelian extensions of k.