A sum-product estimate in fields of prime order

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