A Sum-Product Theorem in Function Fields

Let A be a finite subset of F-q((t(-1))), the field of Laurent series in 1/t over a finite field F-q. We show that, for any epsilon > 0, there exists a constant C dependent only on epsilon and q such that max{vertical bar A + A vertical bar, vertical bar AA vertical bar} >= C vertical bar A vertical bar(6/5-epsilon). In particular, such a result is obtained for the rational function field F-q(t). Identical results are also obtained for finite subsets of the p-adic field Q(p) for any prime p.