Sums of roots of unity vanishing modulo a prime

摘要

In 1976 J.H. Conway and A.J. Jones [1] proved a best possible result to classify the linear equations $ sumlimits_{i=0}^{k-1} a_{i}zeta_{i} = 0 $ satisfied over $ mathbb{Q} $ by a given number k of roots of unity $ zeta_{i} $ . Their result was expressed as an inequality involving k and the common order $ {cal Q} $ of the roots. Here we consider the congruences $ sumlimits_{i=0}^{k-1} a_{i}zeta_{i} equiv 0 , (textrm{mod} , ell) $ , where $ ell $ is a given prime and we obtain the same inequality. As in [1], this yields a classification of such congruences.