Existence problems of primitive polynomials over finite fields

Presern, Mateja (2007) Existence problems of primitive polynomials over finite fields. PhD thesis, University of Glasgow.

Abstract

This thesis concerns existence of primitive polynomials over finite fields with one coefficient
arbitrarily prescribed. It completes the proof of a fundamental conjecture of
Hansen and Mullen (1992), which asserts that, with some explicable general exceptions,
there always exists a primitive polynomial of any degree n over any finite field with an
arbitrary coefficient prescribed. This has been proved whenever n is greater than or equal to 9 or n is less than or equal to 3, but was
unestablished for n = 4, 5, 6 and 8.
In this work, we efficiently prove the remaining cases of the conjecture in a selfcontained
way and with little computation; this is achieved by separately considering
the polynomials with second, third or fourth coefficient prescribed, and in each case developing
methods involving the use of character sums and sieving techniques. When the
characteristic of the field is 2 or 3, we also use p-adic analysis.
In addition to proving the previously unestablished cases of the conjecture, we also
offer shorter and self-contained proof of the conjecture when the first coefficient of the
polynomial is prescibed, and of some other cases where the proof involved a large amount
of computation. For degrees n = 6, 7 and 8 and selected values of m, we also prove the
existence of primitive polynomials with two coefficients prescribed (the constant term and
any other coefficient).

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: The material of Chapters 4 and 5 has been published in mathematical journals, and the material of Chapters 6 and 7 has been accepted for publication. I have not previously submitted any part of this thesis for a degree at any other university.
• Keywords: Finite fields, primitive, primitive elements, primitive polynomials, character sums, Hansen-Mullen primitivity conjecture, p-adic analysis.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Cohen, Prof Stephen D.
• Date of Award: 2007
• Depositing User: Miss Mateja Presern
• Unique ID: glathesis:2007-50
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 11 Jan 2008
• Last Modified: 10 Dec 2012 13:15
• URI: https://theses.gla.ac.uk/id/eprint/50