  We study the Gauss and Jacobi sums from a viewpoint of motives. We exhibit
isomorphisms between Chow motives arising from the Artin-Schreier curve and the
Fermat varieties over a finite field, that can be regarded as (and yield a new
proof of) classically known relations among Gauss and Jacobi sums such as
Davenport-Hasse's multiplication formula. As a key step, we define motivic
analogues of the Gauss and Jacobi sums as algebraic correspondences, and show
that they represent the Frobenius endomorphisms of such motives. This
generalizes Coleman's result for curves. These results are applied to
investigate the group of invertible Chow motives with coefficients in a
cyclotomic field.
