Cyclotomy of Weil sums of binomials

The Well sum W-K,W-d (a) = Sigma(x is an element of) psi (xd + ax )where K is a finite field, psi is an additive character of K, d is coprime to vertical bar K (X)vertical bar, and a is an element of K-X 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 W-K,W-d (a) assumes three distinct values as a runs through K-X. A Galois-theoretic approach, combined with p-divisibility results on Gauss sums, 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. (C) 2015 Elsevier Inc. All rights reserved.