  If $q = p^n$ is a prime power, then a $d$-dimensional \emph{$q$-Butson
Hadamard matrix} $H$ is a $d\times d$ matrix with all entries $q$th roots of
unity such that $HH^* = dI_d$. We use algebraic number theory to prove a strong
constraint on the dimension of a circulant $q$-Butson Hadamard matrix when $d =
p^m$ and then explicitly construct a family of examples in all possible
dimensions. These results relate to the long-standing circulant Hadamard matrix
conjecture in combinatorics.
