Primitive Polynomials of the Form $g(x)+\lambda$ over Finite Fields: Non-Existence Results and Conjectures

Abstract

In this paper, we investigate the existence of primitive polynomials over ${\mathbb{F}}_{q^n}$ whose constant term is a primitive element of ${\mathbb{F}}_{q^n}$. We prove that such polynomials do not exist if $q$ is odd, $q^n\equiv 3(\mathrm{mod}4)$, and the degree $m$ of the polynomial is odd. In particular, the polynomials $f(x)=g(x)+\lambda$, where $g(x)\in {\mathbb{F}}_q[x]$ satisfies $g(0)=0$ and $\lambda \in {\mathbb{F}}_{q^n}$ is primitive, cannot be primitive under the same conditions. Further, for the cubic polynomial $x^3+x^2+x+\lambda$, we establish non-existence results in characteristics $2$ and $3$. These results, in particular, provide counterexamples to previously proposed existence conjectures. We also study the family $x^p+x+\lambda$ over ${\mathbb{F}}_{p^n}$. For an odd prime $p$, we prove that, provided ${\sum }_{i=0}^{n-1}(-1{)}^i{\lambda }^{p^i}\ne 0$, the polynomial $x^p+x+\lambda$ is irreducible over ${\mathbb{F}}_{p^n}$ if and only if $n$ is even. Motivated by this result and supported by computational evidence, we formulate a conjecture concerning the existence of such primitive polynomials.