Primitive free elements of Galois fields
Sophie Huczynska · Enlighten: Theses (The University of Glasgow) · 2002
The key result linking the additive and multiplicative structure of a finite field is the Primitive Normal Basis Theorem; this was established by Lenstra and Schoof in 1987 in a proof which was heavily computational in nature. In this thesis, a new, theoretical proof of the theorem is given, and new estimates (in some cases, exact values) are given for the number of primitive free elements. A natural extension of the Primitive Normal Basis Theorem is to impose additional conditions on the primitive free elements; in particular, we may wish to specify the norm and trace of a primitive free element. The existence of at least one primitive free element of GF(qn) with specified norm and trace was established for n ³ 5 by Cohen in 2000; in this thesis, the result is proved for the most delicate cases, n = 4 and n = 3, thereby completing the general existence theorem.