Primitive Polynomials With Prescribed Second Coefficient

Abstract

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 field with any coefficient arbitrarily prescribed. This has recently been proved whenever n ≥ 9. It is also known to be true when n ≤ 3. We show that there exists a primitive polynomial of any degree n ≥ 4 over any finite field with its second coefficient (i.e

Prescribing coefficients of invariant irreducible polynomials

“…Recently new methods have emerged [14,21,25]. The second result was partially settled by Fan and Han [7,8] and Cohen [4], while the proof was completed by Cohen and Presern [5,6].…”

Problems on Permutation and Irreducible Polynomials Over Finite Fields

“…In 2006, Cohen [19], particularly building on some of the work of Fan-Han [30] on p-adic series, proved there exists a primitive polynomial of degree n ≥ 9 over F q with any one of its coefficients prescribed. The remaining cases of Conjecture A were settled by Cohen-Prešern in [21,22]. Cohen [19] and Cohen-Prešern [21,22] also gave theoretical explanations for the small cases of q, n, missed out in Wan's original proof [75].…”

“…The remaining cases of Conjecture A were settled by Cohen-Prešern in [21,22]. Cohen [19] and Cohen-Prešern [21,22] also gave theoretical explanations for the small cases of q, n, missed out in Wan's original proof [75].…”