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arXiv · 2607.01542

Necessary and sufficient conditions on the order of a finite field $\mathbb{F}_q$ for the easy identification of primitive polynomials of degree 2

Gerardo Vega
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Abstract

We present the necessary and sufficient conditions on the order $q$ of a finite field $\mathbb{F}_q$ such that every irreducible polynomial of the form $x^2+bx+c \in \mathbb{F}_q[x]$, with $b\neq 0$ and $c$ a primitive element of $\mathbb{F}_q$, is a primitive polynomial. As a by-product of this result, we also present a new infinite family of finite fields $\mathbb{F}_q$ for which it is easy, in a different way, to determine when an irreducible polynomial of degree two is primitive.

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Gerardo Vega. 2026-07-01. Necessary and sufficient conditions on the order of a finite field $\mathbb{F}_q$ for the easy identification of primitive polynomials of degree 2. https://arxiv.org/abs/2607.01542

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