Primitive Roots in Cubic Extensions of Finite Fields

N. Katz has shown that the absolute value of sums of the form $$\sum\limits_{b \in {F_q}} X \left( {\theta + b} \right)$$ , F q the finite field of q elements, χ a nontrivial multiplicative character of $${F_{{q^n}}}$$ , and θ a F q -generator of $${F_{{q^n}}}$$ , is bounded from above by $$\left( {n - 1} \right)\sqrt q$$ . We use this result in conjunction with a sieve due to S. Cohen to show the following for n = 3: For any prime power q and any F q -generator θ of $${F_{{q^n}}}$$ , there exists a primitive element of the form aθ + b ∈ $${F_{{q^n}}}$$ for some a, b ∈ F q , a ≠ 0. We discuss an application of these primitive sums in their use as pseudorandom vector generators, and conclude by discussing the harder problem of guaranteeing the existence of such roots when a is forced to be 1.