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.

For an abelian number field K, the discriminant can be obtained from the conductor m of K, the degree of K over Q, and the degrees of extensions, where p runs through the set of primes that divide m, and p α is the greatest power that divides m. In this paper, we give a formula for computing the discriminant of any abelian number field. 2010 AMS Mathematics subject classification. Primary 11R18, 11R29. Keywords and phrases. Cyclotomic fields, abelian number fields, conductors, discriminant.The second author was supported by Consejo Nacional de Ciencia y Tecnología (CONACyT).

It has been shown by Madden that there are only finitely many quadratic extensions of k(x), k a finite field, in which the ideal class group has exponent two and the infinity place of k(x) ramifies. We give a characterization of such fields that allow us to find a full list of all such field extensions for future reference. In doing so we correct some errors in earlier published literature.

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