Primitive polynomials over finite fields of characteristic two
According to a conjecture of \textit{T. Hansen} and \textit{G. L. Mullen} [Math. Comput. 59, 639--643 (1992; Zbl 0770.11053)], for any given element \(a\in\mathbb F_q\) and integers \(n\) and \(m\) there exists a primitive polynomial of degree \(n\) over \(\mathbb F_q\) whose \(m\)th coefficient is equal to \(a\), except for three sporadic cases. In [Proc. Am. Math. Soc. 132, No. 1, 15--31 (2004; Zbl 1047.11119)], the authors showed that the conjecture is true for sufficiently large \(q\) (depending on \(n\)) and \(m\not=\frac{n+1}{2}\) or \(m\not=\frac{n}{2},\frac{n}{2}+1\). Now they prove the existence of primitive polynomials with an arbitrary prescribed coefficient over fields of characteristic two for odd values of \(n\geq7\) whatever the size of \(q\). The existence problem of primitive polynomials over finite fields is translated into the existence problem of primitive element solutions of systems of trace equations over Galois rings. Then character sum methods and Cohen's sieve yield conditions for the solvability of such systems.
primitive polynomials over finite fields
scientific article; zbMATH DE number 2101931