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PRIMITIVE POLYNOMIALS WITH PRESCRIBED SECOND COEFFICIENT

Published online by Cambridge University Press:  23 August 2006

STEPHEN D. COHEN
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland e-mail: sdc@maths.gla.ac.uk, mp@maths.gla.ac.uk
MATEJA PREšERN
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland e-mail: sdc@maths.gla.ac.uk, mp@maths.gla.ac.uk
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Abstract

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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 $n \geq 9$. It is also known to be true when $n \leq 3$. We show that there exists a primitive polynomial of any degree $n\geq 4$ over any finite field with its second coefficient (i.e., that of $x^{n-2}$) arbitrarily prescribed. In particular, this establishes the HMPC when $n=4$. The lone exception is the absence of a primitive polynomial of the form $x^4+a_1x^3 +x^2+a_3x+1$ over the binary field. For $n \geq 6$ we prove a stronger result, namely that the primitive polynomial may also have its constant term prescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs.

Type
Research Article
Copyright
2006 Glasgow Mathematical Journal Trust