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. For the general case, we use the fact that the extended genus fields of a composition of two cyclotomic extensions of a global rational function field is the same as the composition of their corresponding extended genus fields. In the main result of the paper, we give the extended genus field of finite abelian extensions of a global rational function field explicitly in terms of the field and extended genus field of its “cyclotomic projection”.