Primitive Polynomials Over Finite Fields

Hansen, Tom; Mullen, Gary L.

doi:10.2307/2153081

Cited by 27 publications

(46 citation statements)

References 3 publications

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“…Both results had been conjectured by Hansen and Mullen [16]. Theorem 1.1 was initially proved for q > 19 or n ≥ 36 by Wan [26], while Han and Mullen [15] verified the remaining cases by computer search.…”

Section: Introductionmentioning

confidence: 79%

Prescribing coefficients of invariant irreducible polynomials

Kapetanakis

2017

Journal of Number Theory

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Let F q be the finite field of q elements. We define an action of PGL(2, q) on F q [X] and study the distribution of the irreducible polynomials that remain invariant under this action for lower-triangular matrices. As a result, we describe the possible values of the coefficients of such polynomials and prove that, with a small finite number of possible exceptions, there exist polynomials of given degree with prescribed high-degree coefficients.

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Section: Introductionmentioning

confidence: 79%

Prescribing coefficients of invariant irreducible polynomials

Kapetanakis

2017

Journal of Number Theory

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“…The methods of Hansen and Mullen [12] provide the starting point for our search. Because of the availability of software programs used in [12], we have used the following strategy in searching for a primitive normal polynomial over Fp, even though some of the methods discussed in § 1 at least theoretically provide faster algorithms.…”

Section: Tablesmentioning

confidence: 99%

“…Because of the availability of software programs used in [12], we have used the following strategy in searching for a primitive normal polynomial over Fp, even though some of the methods discussed in § 1 at least theoretically provide faster algorithms. We first locate a primitive polynomial of degree n over Fp , as in [12], using Theorem 1.…”

Section: Tablesmentioning

confidence: 99%

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Primitive Normal Polynomials Over Finite Fields

Morgan

Mullen²

1994

Mathematics of Computation

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Abstract.In this note we significantly extend the range of published tables of primitive normal polynomials over finite fields. For each p" < 1050 with P < 97, we provide a primitive normal polynomial of degree n over Fp. Moreover, each polynomial has the minimal number of nonzero coefficients among all primitive normal polynomials of degree n over Fp. The roots of such a polynomial generate a primitive normal basis of Fpn over Fp , and so are of importance in many computational problems. We also raise several conjectures concerning the distribution of such primitive normal polynomials, including a refinement of the primitive normal basis theorem.

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“…One of the basic problems in computational number theory is to investigate the distribution of the coefficients of primitive polynomials; that is, whether there exists a primitive polynomial with one coefficient or several coefficients prescribed in advance. Based on various tables, Hansen and Mullen [10] proposed the following conjecture about the distribution of primitive polynomials with one coefficient prescribed.…”

Section: Introductionmentioning

confidence: 99%