Cyclotomy of Weil sums of binomials

The Weil sum WK,d(a)=∑x∈Kψ(xd+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, 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.