Primitive free elements of Galois fields

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

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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
Funder's Name: UNSPECIFIED
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: http://theses.gla.ac.uk/id/eprint/5533

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