  In this paper we first obtain the genus field of a finite abelian non-Kummer
$l$--extension of a global rational function field. Then, using 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 and our previous results, we deduce the general
expression of the genus field of a finite abelian extension of a global
rational function field.
