Cyclotomy of Weil Sums of Binomials

The Weil sum WK,d(a) = ∑ x∈K ψ(x d + ax) where K is a finite field, ψ is an additive character of K, d is coprime to |K|, and a ∈ K arises often in number-theoretic calculations, and in applications to finite geometry, cryptography, digital sequence design, and coding theory. Researchers are especially interested in the case where WK,d(a) assumes three distinct values as a runs through K. A Galois-theoretic approach is used here to prove a variety of new results that constrain which fields K and exponents d support three-valued Weil sums, and restrict the values that such Weil sums may assume.