Consider
a finite field F=GF(q) and its degree n extension E=GF(q^n). The
Primitive Normal Basis Theorem (proved in 1987) guarantees the
existence of an element of E which is simultaneously primitive and free
over F; in other words, an element which is simutaneously a
multiplicative and additive generator for the field E. Subsequently,
there has been interest in the existence of primitive free elements
with various extra properties; answers to such existence questions have
combinatorial applications, as well as being interesting in their own
right. In this talk, I will discuss a framework for answering such
questions, and present a new result of this kind, the "Strong Primitive
Normal Basis Theorem". (This is joint work with S.D.Cohen.)