This Talk is in
the area of algebraic number theory. More specifically, we use explicit methods
to determine the "roots of unity modulo M", namely, the solutions of
equation x^n =1 in the ring O_K / M, where O_K is the ring of integers of a number
field K. and M is an (integral) ideal of O_K.
The solutions are then applied to generalize a theorem of Bauer, counting the
number of solutions to Pell equations modulo rational prime p and generalizing
a theorem of Wolstenholme.