The Primitive Normal Basis Theorem – Without a Computer

An element α of the extension E of degree n over the finite field F = GF (q) is called free over F if {α, α q , . . . , α q n−1 } is a (normal) basis of E/F . The primitive normal basis theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F , there exists an element α ∈ E such that α is simultaneously primitive (i.e., generates the multiplicative group of E) and free over F . In this paper we prove the following strengthening of this theorem: aside from five specific extensions E/F , there exists an element α ∈ E such that both α and α −1 are simultaneously primitive and free over F .

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“…and less computational approach realised in [5] was the introduction of sieving techniques (cf. Section 4, below).…”

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confidence: 99%

The strong primitive normal basis theorem

Cohen

Huczynska

2010

Acta Arith.

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“…We string the 82 pairs (a, c), one example for each primitive quartic (2,14), (3,5), (4,35), (5,14), (6,24), (7,20), (8,34), (9,14), (10, 2), (11,2), (12,8), (13,15), (14,6), (15,14), (16,80), (17,6), (18,32), (19,19), (20,43), (21,15), (22, 32), (23,32), (24,8), (25,62) In summary, the above examples suffice to complete the proof (for odd q) of Theorem 1.2 for n = 4, m = 2 and a = 0.…”

Section: The Odd Non-zero Problemmentioning

confidence: 99%

“…Then there exists a primitive polynomial of degree n over ‫ކ‬ q with its coefficient of x 2 equal to a. In particular, the HMPC is established for (n, m) = (6, 4), (7,5) or (8,6). …”

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Primitive Polynomials With Prescribed Second Coefficient

Cohen

Presern

2006

Glasgow Math. J.

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Abstract. The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite field with any coefficient arbitrarily prescribed. This has recently been proved whenever n ≥ 9. It is also known to be true when n ≤ 3. We show that there exists a primitive polynomial of any degree n ≥ 4 over any finite field with its second coefficient (i.e

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“…Existence of such an element for every extension was demonstrated by Lenstra and Schoof [LeSc] (completing work by Carlitz ([Ca1], [Ca2]) and Davenport [Da]). A computer-free proof of the primitive normal basis theorem is given in [CoHu1].…”

Section: Introductionmentioning

confidence: 99%