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)

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