On the Existence of Primitive Polynomials $f(x)=g(x)+ \lambda$ over Finite Fields

Abstract

In this paper, we study the existence of primitive polynomials of the form $f(x)=g(x)+\lambda$, where $g(x)\in\mathbb{F}_q[x]$ satisfies $g(0)=0$ and $\lambda$ is a primitive element of $\mathbb{F}_{q^n}$. This problem is motivated by Conjectures 4.1 and 9.1 of \cite{TSR}. Conjecture 4.1 predicts that, for every $m,n\geq 2$, there exists a primitive polynomial of degree $m$ over $\mathbb{F}_{q^n}$ of the form $f(x)=g(x)+\lambda$, where $g(0)=0$ and $\lambda$ is a primitive element of $\mathbb{F}_{q^n}$. Conjecture 9.1 further asserts that, for $m=3$ and $n=2$, the polynomial $x^3+x^2+x+\alpha$ is primitive over $\mathbb{F}_{q^2}$ for every prime power $q$, where $\alpha$ is a generator of the multiplicative group $\mathbb{F}_{q^2}^{*}$. We show that these conjectures are false in general by constructing explicit counterexamples over suitable finite fields. On the other hand, assuming that the characteristic of $\mathbb{F}_q$ does not divide $m$, we derive a sufficient condition for the existence of primitive polynomials of the prescribed form. As a consequence, we prove that for all $m\geq 3$ and $n\geq 2$, the conjectured polynomials exist over sufficiently large finite fields whose characteristic does not divide $m$.