PRIMITIVE POLYNOMIALS WITH PRESCRIBED SECOND COEFFICIENT

COHEN, STEPHEN D. ; PREšERN, MATEJA

Glasgow mathematical journal, Tome 48 (2006) no. 2, pp. 281-307

Résumé

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.

Détail

DOI : 10.1017/S0017089506003077

Mots-clés : 11T06, 11T30, 11T24, 11L40, 11S85