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arXiv · 2501.03516

The Multiple Equal-Difference Structure of Cyclotomic Cosets

Li Zhu·Juncheng Zhou·Jinle Liu·Hongfeng Wu
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Abstract

In this paper we introduce the definition of equal-difference cyclotomic coset, and prove that in general any cyclotomic coset can be decomposed into a disjoint union of equal-difference subsets. Among the equal-difference decompositions of a cyclotomic coset, an important class consists of those in the form of cyclotomic decompositions, called the multiple equal-difference representations of the coset. There is an equivalent correspondence between the multiple equal-difference representations of $q$-cyclotomic cosets modulo $n$ and the irreducible factorizations of $X^{n}-1$ in binomial form over finite extension fields of $\mathbb{F}_{q}$. We give an explicit characterization of the multiple equal-difference representations of any $q$-cyclotomic coset modulo $n$, through which a criterion for $X^{n}-1$ factoring into irreducible binomials is obtained. In addition, we present an algorithm to simplify the computation of the leaders of cyclotomic cosets.

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Li Zhu, Juncheng Zhou, Jinle Liu, Hongfeng Wu. 2025-01-07. The Multiple Equal-Difference Structure of Cyclotomic Cosets. https://arxiv.org/abs/2501.03516

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