Rational points on Fermat curves over finite fields

Let ( u, v, w) be the F-qi-rational point on the Fermat curve u(q-1) vertical bar v(q-1) vertical bar w(q-1) = 0 with 3|(q(i) -1). It has recently been proved that if i is an element of{1, 2, 3} then each uvw is a cube in F qi. It is natural to wonder whether there is a generalization to i >= 4. In this paper, we show that the result cannot be extended to i >= 4 in general and conjecture that each uvw is a cube in F-q(i) if and only if i is an element of{1, 2, 3}.