  We give an explicit description of the F_{q^i}-rational points on the Fermat
curve u^{q-1}+v^{q-1}+w^{q-1}=0 for each i=1,2,3. As a consequence, we observe
that for any such point (u,v,w), the product uvw is a cube in F_{q^i}. We also
describe the F_{q^2}-rational points on the Fermat surface
u^{q-1}+v^{q-1}+w^{q-1}+x^{q-1}=0, and show that the product of the coordinates
of any such points is a square.
