The Optimal Coefficients-Based Criterion for Primitive Quadratic Polynomials over Finite Fields

Abstract

Let ${\mathbb{F}}_q$ be a finite field and consider quadratic polynomials $f(X)=X^2+bX+c\in {\mathbb{F}}_q[X]$ with primitive constant term $c$. We construct an optimal coefficients-based determining polynomial for primitive quadratic polynomials over every finite field. More precisely, for every primitive $c\in {\mathbb{F}}_q$, its specialization is the unique monic square-free polynomial whose roots are exactly the coefficients $b$ for which $X^2+bX+c$ is primitive. The construction is based on Lucas polynomials and Lucas atoms over finite fields. In odd characteristic, the optimal determining polynomial is the $(q+1)$-st Lucas atom. In characteristic $2$, this Lucas atom has multiplicity $2$ in the variable $B$, and the optimal polynomial is obtained by removing this multiplicity through the inverse Frobenius over $A={\mathbb{F}}_q[C]/({\Phi }_{q-1}(C))$. We prove that the determining polynomial is unique for each fixed primitive constant term and, globally, unique as an element of $A[B]$. We also give equivalent criteria involving only recursively computable Lucas polynomials and, as an application, a first-zero coefficient description of the binomial order and order of an irreducible quadratic polynomial.