有限域上不可约二次多项式的二项式阶与本原性

中文摘要 AI 辅助

本文研究有限域上本原二次多项式的二项式阶与系数判据。首先，引入与不可约二次多项式关联的Lucas序列，证明其首零项同时决定二项式阶与多项式的阶，从而证实Vega的一个猜想。其次，通过计算具有固定本原常数项的不可约与本原二次多项式的数量，完全分类Vega的系数判据$b^2\neq2c$成立的有限域，解答Vega提出的另一个公开问题。最后，对满足$q+1=2^m\pi$（$\pi$为奇素数）的有限域，基于递归定义的例外多项式族，建立本原二次多项式的统一系数判据，拓展了此前$q+1=2\pi$和$q+1=4\pi$的已知判据。

英文摘要

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 $A=\mathbb F_q[C]/(Φ_{q-1}(C))$. 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.

发表机构

• College of Science, North China University of Technology（华北理工大学理学院）
• School of Mathematical Sciences, Guizhou Normal University（贵州师范大学数学科学学院）

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