Let [[zeta].sub.5] be a primitive 5th

root of unity and put

By Theorem 2.7, each [[lambda].sub.i] is a primitive kth

root of unity for some k | n.

Any complex number with absolute value 1 is arbitrarily close to a

root of unity and any symbol containing a

root of unity is trivial since, for example, {[zeta], b} = (1/m){[[zeta].sup.m], b} = 0 if [zeta] is an mth

root of unity.

An n-th

root of unity, where n is a positive integer (i.e., n = 1, 2, 3, ....), is a number z satisfying the equation

(0) For every

root of unity [zeta] there is a [theta]-function [theta] [sub.[zeta]](q) such that the difference f(q) -[theta] [sub.[zeta]](q)is bounded as q [right arrow] radially.

Let [epsilon] = [e.sup.[2[pi]i/n]] be the n-th

root of unity and consider the polynomial [X.sup.n] - 1.

A triple (X, X(q), C) consisting of a finite set X, a polynomial X(q) [member of] N[q] satisfying X(1) = [absolute value of X], and a cyclic group C acting on X exhibits the cyclic sieving phenomenon if, for every c [member of] C, if w is a primitive

root of unity of the same multiplicative order as c, then

This notion has been generalized to the notion of graded q-differential algebra, where q is a primitive Nth

root of unity (see papers [7,8,10]).

A complex number [omega] is said to be an n-th

root of unity if [omega.sup.n] = 1.

Similarly, when p [member of] [I.sub.p] and p is not a

root of unity then we can write [I.sub.p] = [I.sup.+.sub.p] [??] [I.sup.- .sub.p], where [mathematical expression not reproducible].