Narrow Results

We give a construction of the genus field for Kummer ${\ell }^n$-cyclic extensions of rational congruence function fields, where $\ell$ is 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 $\ell$-cyclic extension. Finally, we study the extension ${(K_1{K}_2)}_{𝔤𝔢}/{(K_1)}_{𝔤𝔢}{(K_2)}_{𝔤𝔢}$, for $K_1$, $K_2$ abelian extensions of $k$.

We give a construction of genus fields for Kummer cyclic l-extensions of rational congruence function fields, l a prime number. First we find this genus field for a field contained in a cyclotomic function field using Leopoldt's construction by means of Dirichlet characters and the Hilbert class field defined by Rosen. The general case follows from this. This generalizes the result obtained by Peng for a cyclic extension of degree l.