Primitive Polynomials of the Form $g(x)+λ$ over Finite Fields: Non-Existence Results and Conjectures
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, $q^n\equiv3\pmod{4}$, and the degree $m$ of the polynomial is odd. In particular, the polynomials $f(x)=g(x)+λ$, where $g(x)\in\mathbb{F}_q[x]$ satisfies $g(0)=0$ and $λ\in\mathbb{F}_{q^n}$ is primitive, cannot be primitive under the same conditions. Further, for the cubic polynomial $x^3+x^2+x+λ$, 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 $x^p+x+λ$ over $\mathbb{F}_{p^n}$. For an odd prime $p$, we prove that, provided $\sum_{i=0}^{n-1}(-1)^iλ^{p^i}\neq0$, the polynomial $x^p+x+λ$ 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 conjectures concerning the existence of such primitive polynomials, including the stronger assertion that $x^p+x+λ$ is primitive for every primitive $λ\in\mathbb{F}_{p^2}$.