  Can any element in a sufficiently large finite field be represented as a sum
of two $d$th powers in the field? In this article, we recount some of the
history of this problem, touching on cyclotomy, Fermat's last theorem, and
diagonal equations. Then, we offer two proofs, one new and elementary, and the
other more classical, based on Fourier analysis and an application of a
nontrivial estimate from the theory of finite fields. In context and
juxtaposition, each will have its merits.
