  For a number field $K$ with ring of integers ${\mathcal O}_K$, we prove an
analogue over finite rings of the form ${\mathcal O}_K/{\mathcal P}^m$ of the
Fundamental Theorem on the Fourier transform of a relative invariant of
prehomogeneous vector spaces, where ${\mathcal P}$ is a big enough prime ideal
of ${\mathcal O}_K$ and $m>1$. In the appendix, F. Sato gives an application of
the Theorems A, B and the Theorems A, B, C in J. Denef and A. Gyoja
[\emph{Character sums associated to prehomogeneous vector spaces}, Compos.
Math., \textbf{113} (1998) 237--346] to the functional equation of
$L$-functions of Dirichlet type associated with prehomogeneous vector spaces.
