  Cyclotomy, the study of cyclotomic classes and cyclotomic numbers, is an area
of number theory first studied by Gauss. It has natural applications in
discrete mathematics and information theory. Despite this long history, there
are significant limitations to what is known explicitly about cyclotomic
numbers, which limits the use of cyclotomy in applications. The main explicit
tool available is that of uniform cyclotomy, introduced by Baumert, Mills and
Ward in 1982. In this paper, we present an extension of uniform cyclotomy which
gives a direct method for evaluating all cyclotomic numbers over $GF(q^n)$ of
order dividing $(q^n-1)/(q-1)$, for any prime power $q$ and $n \geq 2$, which
does not use character theory nor direct calculation in the field. This allows
the straightforward evaluation of many cyclotomic numbers for which other
methods are unknown or impractical. Our methods exploit connections between
cyclotomy, Singer difference sets and finite geometry.
