Extended genus fields of abelian extensions of rational function fields

In this paper we obtain the extended genus field of a finite abelian extension of a global rational function field. We first study the case of a cyclic extension of prime power degree. Next, we use that the extended genus fields of a composite of two cyclotomic extensions of a global rational function field is equal to the composite of their respective extended genus fields, to obtain our main result. This result is that the extended genus field of a general finite abelian extension of a global rational function field, is given explicitly in terms of the field and of the extended genus field of its "cyclotomic projection".