Since F\{0} consists of invertible elements,
the multiplicative group F* of F has order q - 1.
If x = 0, then clearly xq = x.
Suppose, therefore, x 
0.
Then x belongs to F*
and so a corollary to Lagrange's Theorem yields
xq - 1 = 1. The required equation
follows when we multiply both sides by x.