Cyclotomy for non-squarefree modul I

Abstract

Kummer considered cyclotomy for a composite modulus n and defined the periods by

$$n_k = \sum\limits_{v = 0}^{f - 1} {\zeta _n^{kq^\upsilon } } k\not \equiv q^j (\bmod n),0 < j < \phi (n)$$

where ζ_n=exp(2πi/n) and f=t(n) is the order of q modulo n. He stated that, except when all the η's vanish, they satisfy an irreducible monic period equation with integer coefficients of degree ϕ(n)/t(n). Soon thereafter Fuchs gave a necessary and sufficient condition for the vanishing of the η's, namely: n_k=0 if and only if t(n)=pt(n/p) for some prime p dividing n. A modern proof for a generalized period was recently given by R.J. Evans.

The present paper examines the differences between Kummer's periods for composite n, when n is not squarefree, and the classical Gaussian periods for n a prime. Besides the fact that for a given non-squarefree n there exist values of q for which the η's vanish, the cyclotomy may no longer be unique. These and other distinctions are illustrated by examples.

Just as the Gaussian periods give rise to families of difference sets, the Kummer periods can be used to explain some of the known Singer difference sets arising from finite projective geometry. It is hoped that the further study of Kummer periods may bring to light a new family of difference sets.

Cyclotomy is to be regarded not as an incidental application, but as the natural and inherent centre and core of the arithmetic of the future. J.J. Sylvester, 1879.