Primitive free elements of Galois fields

Huczynska, Sophie (2002) Primitive free elements of Galois fields. PhD thesis, University of Glasgow.

Abstract

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.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Cohen, Prof. S.D. and Odoni, R.W.K.
• Date of Award: 2002
• Depositing User: Mrs Marie Cairney
• Unique ID: glathesis:2002-5533
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 24 Sep 2014 08:54
• Last Modified: 24 Sep 2014 15:05
• URI: https://theses.gla.ac.uk/id/eprint/5533