  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.
