Genus fields of Kummer $\ell^n$-cyclic extensions

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 $(K_1K_2)_{\frak{ge}}/(K_1)_{\frak{ge}}(K_2)_{\frak{ge}}$, for $K_1$, $K_2$ abelian extensions of $k$.