线性化多项式核的刻画

Characterizations of Kernel of Linearized Polynomials

摘要: 本文在总结相关文献的基础上，整理了${\mathbb{F}}_{q^n}$ 上的线性化多项式核的多种刻画方式。首先，总结了${\mathbb{F}}_q$ 上线性化多项式代数$\mathbb{L}\left({\mathbb{F}}_q\right)$ 的循环矩阵刻画。接着在回顾了线性化多项式的“迹表示”后，通过“迹表示”及初等方法证明了Dickson关于线性化置换多项式的知名判定法则，并再次得到了${\mathbb{F}}_{q^n}$ 上的线性化多项式代数与Dickson矩阵代数间的同构关系。

Abstract: In this paper, we summarize some characterizations of the kernel of linearized polynomials over ${\mathbb{F}}_{q^n}$ after reviewing related articles. Firstly, circulant matrices characterization of algebra $\mathbb{L}\left({\mathbb{F}}_q\right)$ over ${\mathbb{F}}_q$ are summed up. Then, after reviewing the “trace representations” of linearized polynomials, we prove Dickson’s well-known decision rule for permutation linearized polynomials by elementary methods and “trace representations”, then obtain the isomorphism between linearized polynomials algebra and Dickson matrices algebra over ${\mathbb{F}}_{q^n}$ again.