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

Rational points on some Fermat curves and surfaces over finite fields