A characterization and an explicit description of all primitive polynomials of degree two over finite fields

TL;DR

This paper improves the characterization of degree two polynomials over finite fields and provides a complete explicit description of all primitive polynomials of this degree, aiding their identification and application.

Contribution

It offers an improved characterization and a simple, explicit description of all primitive degree two polynomials over finite fields, advancing understanding in finite field theory.

Findings

Enhanced characterization of degree two polynomials

Explicit description of all primitive degree two polynomials

Simplified criteria for identifying primitive polynomials

Abstract

For polynomials of degree two over finite fields, we present an improvement of Fitzgerald's characterization (Finite Fields Appl. 9(1):117-121, 2003). We then use this new characterization to obtain an explicit, complete, and simple description of all primitive polynomials of degree two over finite fields.