Primitive polynomials with prescribed second coefficient

Cohen, S.D. and Presern, M. (2006) Primitive polynomials with prescribed second coefficient. Glasgow Mathematical Journal, 48(2), pp. 281-307. (doi:10.1017/S0017089506003077)

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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 fieldwith any coefficient arbitrarily prescribed. This has recently been provedwhenever n ≥ 9. It is also known to be truewhen n ≤ 3.We showthat there exists a primitive polynomial of any degree n ≥ 4 over any finite field with its second coefficient (i.e., that of xn−2) arbitrarily prescribed. In particular, this establishes the HMPC when n = 4. The lone exception is the absence of a primitive polynomial of the form x4 + a1x3 + x2 + a3x + 1 over the binary field. For n ≥ 6 we prove a stronger result, namely that the primitive polynomialmay also have its constant termprescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Cohen, Professor Stephen
Authors: Cohen, S.D., and Presern, M.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Glasgow Mathematical Journal
Publisher:Cambridge University Press
ISSN:0017-0895
ISSN (Online):1469-509X
Published Online:23 August 2006
Copyright Holders:Copyright © 2006 Glasgow Mathematical Journal Trust
First Published:First published in Glasgow Mathematical Journal 48(2):281-307
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher

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