The Hansen-Mullen primitivity conjecture: completion of proof

Summary

Abstract. This paper completes an efficient proof of the Hansen-Mullen Primitivity Conjecture (HMPC) when n = 5, 6, 7 or 8. The 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 or n ≤ 4. We show that there exists a primitive polynomial of any degree n ≥ 5 over any finite field with third coefficient, i.e., the coefficient of xⁿ⁻³, arbitrarily prescribed. This completes the HMPC when n = 5 or 6. For n ≥ 7 we prove a stronger result, namely that the primitive polynomial may also have its constant term prescribed. This implies further cases of the HMPC and completes the HMPC when n = 7. We also show that there exists a primitive polynomial of degree n ≥ 8 over any finite field with the coefficient of xⁿ⁻⁴ arbitrarily prescribed, and this completes the HMPC when n = 8. A feature of the method, when the cardinality of the field is 2 or 3, is that 2-adic and 3-adic analysis is required for the proofs. The article is intended to provide the reader with an overview of the general approach to the solution of the HMPC without the weight of detail involved in unravelling the situation of arbitrary degree.