On vanishing sums of roots of unity

An unsolved problem in number theory asked the following: For a given natur al number In, what are the possible integers n for which there exist mth ro ots of unity alpha(1),..., alpha(n) is an element of C such that alpha(1) ... + alpha(n) = 0? We show in this paper that the set of all possible n's is exactly the collection of N-combinations of the prime divisors of m, wh ere N denotes the set of all non-negative integers. The proof is long and i nvolves a subtle analysis of minimal vanishing sums of mth roots of unity, couched in the setting of integral group rings of finite cyclic groups. Our techniques also recovered with ease some of the classical results on vanis hing sums of roots of unity, such as those of Redei, de Bruijn, and Schoenb erg. (C) 2000 Academic Press.