  We propose a normal form for single-qudit gates composed of Clifford and
$T$-gates for qudits of odd prime dimension $p\geq 5$. We prove that any
single-qudit Clifford+$T$ operator can be re-expressed in this normal form in
polynomial time. We also provide strong numerical evidence that this normal
form is unique. Assuming uniqueness, we are able to use this normal form to
provide an algorithm for exact synthesis of any single-qudit Clifford+$T$
operator with minimal $T$-count.
