### Existence problems for generators of finite fields

Stephen D Cohen
*Department of Mathematics, University of
Glasgow*

The multiplicative group of a finite field *E* is
cyclic: a generator is a *primitive element*. If *E* is the
extension *GF(q^n)* of *F:=GF(q)*, then, additively, viewed as
an *FG*-module (where *G=Gal(E/F)*), *E* is also cyclic: a
generator is a *free element of E over F*. Classically,
this is the *normal basis theorem* whereby *E* possesses an
*F*-basis of the form *{w, w^q, w^{q^2},..., w^{q^{n-1}}}*. The
notions of a *primitive* and a *free* element are
largely independent but the existence of a primitive element of
*E*, free over *F* was established by Lenstra and Schoof in 1987
(building on earlier work of Carlitz and of Davenport) using Gauss
sums to tie together the multiplicative and additive aspects.
We shall discuss further existence problems with a view to
obtaining results displaying a measure of completeness. For
example, a recent result is that, for all *q* and all *n >= 5*,
there exists a primitive element of *E* free over *F* with
arbitrary (non-zero) *E/F*- trace and arbitrary (primitive)
*E/F*-norm. The proof uses a refined expression in terms of Gauss
sums, and a sieving technique with respect to the orders of both
the multiplicative and additive characters involved.