After a brief review of the notion of a full set of mutually unbiased bases in an N-dimensional Hilbert space, we summarize the work of Wootters and Fields (W K Wootters and B C Fields, Ann. Phys.191, 363 (1989)) which gives an explicit construction for such bases for the case N=p^r, where p is a prime. Further, we show how, by exploiting certain freedom in the Wootters-Fields construction, the task of explicitly writing down such bases can be simplified for the case when p is an odd prime. In particular, we express the results entirely in terms of the character vectors of the cyclic group G of order p. We also analyse the connection between mutually unbiased bases and the representations of G.