Let A be a matrix of finite order n and finite degree d, whose characteristic roots are certain n
th roots of unity a1, a2…, a
. We wish to prove a congruence (6) between the traces (tr) of certain powers of A, which is suggested by two somewhat simpler congruences (1) and (3). First, if tr (A) is a rational integer, it is easy to establish the familiar congruence
even though tr(A
) may not itself be rational.