On Jacobi sums and difference sets

Abstract

Let L be the cyclotomic field of the e-th roots of unity where e is even, and p be a prime of the form 1+ ef. The Jacobi sums for the e-th power residue characters of p, divisible by a prime ideal divisor P of p in L, together with the torsion group W of L forms a multiplicative group J. It is shown in this paper that J is embedded into a group J 0=W×A , where A is a free Abelian group of rank φ(e)/2, in a manner quite independent of p. On the other hand, a necessary and sufficient condition for a difference set over a cyclic group of p elements which has a multiplier group of index e is obtained. Combining these two theorems, we can determine all the cyclic difference set with e≤12.