The Cassels heights of cyclotomic integers

Abstract

We study the set C of mean square values of the moduli of the conjugates of all nonzero cyclotomic integers beta. For its kth derived set C-(k), we show that C-(k) = (k + 1)C (k >= 0), so that also C-(k) + C-(l) = C( k+l+1) (k, l >= 0). Furthermore, we describe precisely the restricted set C-p where the beta are confined to the ring Z[omega(p)], where p is an odd prime and omega(p) is a primitive pth root of unity. In order to do this, we prove that both of the quadratic polynomials a(2) + ab + b(2) + c(2) + a + b +c and a(2) + b(2) + c(2) + ab + bc + ca + a + b + c are universal.