Existence problems of primitive polynomials over finite fields

QA Mathematics

Presern, Mateja ; Cohen, Stephen D.

University:University of Glasgow

Department:School of Mathematics and Statistics

关键词: Finite fields, primitive, primitive elements, primitive polynomials, character sums, Hansen-Mullen primitivity conjecture, p-adic analysis.;   

Others  :  http://theses.gla.ac.uk/50/1/2007presernphd.pdf

来源: University of Glasgow

【 摘 要 】

This thesis concerns existence of primitive polynomials over finite fields with one coefficientarbitrarily prescribed. It completes the proof of a fundamental conjecture ofHansen 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 anarbitrary 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 wasunestablished for n = 4, 5, 6 and 8.In this work, we efficiently prove the remaining cases of the conjecture in a selfcontainedway and with little computation; this is achieved by separately consideringthe polynomials with second, third or fourth coefficient prescribed, and in each case developingmethods involving the use of character sums and sieving techniques. When thecharacteristic 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 alsooffer shorter and self-contained proof of the conjecture when the first coefficient of thepolynomial is prescibed, and of some other cases where the proof involved a large amountof computation. For degrees n = 6, 7 and 8 and selected values of m, we also prove theexistence of primitive polynomials with two coefficients prescribed (the constant term andany other coefficient).