  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".
