Cyclotomy of Weil Sums of Binomials

The Weil sum $W_{K,d}(a)=\sum_{x \in K} \psi(x^d + a x)$ where $K$ is a finite field, $\psi$ is an additive character of $K$, $d$ is coprime to $|K^\times|$, and $a \in K^\times$ 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 $W_{K,d}(a)$ assumes three distinct values as $a$ runs through $K^\times$. 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.