A recprocity relation between some cyclotomic numbers
by
Charles Helou
Penn State University

We study a reciprocity relation between the factorization of 1-(1-\zeta_p)^q in Q(\zeta_p) and that of 1-(1-\zeta_q)^p in Q(\zeta_q), where p, q are prime numbers and \zeta_p, \zeta_q are primitive p-th, q-th roots of unity respectively. In particular, one of these cyclotomic numbers is prime if and only if the other is. We also obtain formulas for their norms for some small values of p or q.

Date received: March 10, 2000