Primitive elements of finite fields Fqr avoiding affine hyperplanes for q = 4 and q = 5

Abstract

For a finite field $F_{q^r}$ with fixed q and r sufficiently large, we prove the existence of a primitive element outside of a set of r many affine hyperplanes for $q=4$ and $q=5$. This complements earlier results by Fernandes and Reis for $q\ge 7$. For $q=3$ the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For $q=2$ the set consists only of a single element, and such a result is thus not possible.