  Let q be a prime, A be a subset of a finite field $F=\Bbb Z/q\Bbb Z$,
$|A|<\sqrt{|F|}$. We prove the estimate $\max(|A+A|,|A\cdot A|)\ge
c|A|^{1+\epsilon}$ for some $\epsilon>0$ and c>0. This extends the result of J.
Bourgain, N. Katz, and T. Tao.
