Character Sums, Primitive Elements, and Powers in Finite Fields

Abstract

Consider an extension field Fqm=Fq(α) of the finite field Fq. Davenport proved that the set Fq+α contains at least one primitive element of Fqm if q is sufficiently large with respect to m. This result is extended to certain subsets of Fq+α of cardinality at least of the order of magnitude O(q1/2+ε). The proof is based on a new bound for incomplete character sums. Moreover, a new bound for the longest sequence of consecutive powers in Fqm is deduced.