Sum uniform subsets of the integers modulo p and an application to finite fields

Abstract

We show that, if p ≠ 3 is an odd prime satisfying p ≢ 5 ( mod 8 ) , then each nonzero element of GF ( p ) can be written as a sum of distinct quadratic residues in the same number of ways, N say, and that the number of ways of writing 0 as a sum of distinct quadratic residues is N + ( 2 p ) , where ( 2 p ) is the Legendre symbol. We actually prove a more general result on sum uniform subgroups of GF ( p ) ∗ , which holds for any odd prime p ≠ 3 . These results are applied to the problem of determining subgroups H of the multiplicative group of a finite field, with the property that 1 + h is a non-square of the field, for all h ∈ H .