  In this paper we obtain the genus field of a general Kummer extension of a
global rational function field. We study first the case of a general Kummer
extension of degree a power of a prime. Then we prove that the genus field of a
composite of two abelian extensions of a global rational function field with
relatively prime degrees is equal to the composite of their respective genus
fields. Our main result, the genus of a general Kummer extension of a global
rational function field, is a direct consequence of this fact.
