  For a cyclic Kummer extension $K$ of a rational function field $k$ is
considered, via class field theory, the extended Hilbert class field $K_H^+$ of
$K$ and the corresponding extended genus field $K_g^+$ of $K$ over $k$, along
the lines of the definitions of R. Clement for such extensions of prime degree.
We obtain $K_g^+$ explicitly. Also, we use cohomology to determine the number
of ambiguous classes and obtain a reciprocity law for $K/k$. Finally, we
present a necessary and sufficient condition for a prime of $K$ to decompose
fully in $K_g^+$.
