A Note on Factorization and the Number of Irreducible Factors of xn − λ over Finite Fields

Liu, Jinle; Wu, Hongfeng

doi:10.3390/math13030473

Open AccessArticle

A Note on Factorization and the Number of Irreducible Factors of xⁿ − λ over Finite Fields

by

Jinle Liu

and

Hongfeng Wu

^*

College of Science, North China University of Technology, Beijing 100144, China

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(3), 473; https://doi.org/10.3390/math13030473

Submission received: 26 November 2024 / Revised: 24 December 2024 / Accepted: 25 December 2024 / Published: 31 January 2025

(This article belongs to the Section A: Algebra and Logic)

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Abstract

Let

F_{q}

be a finite field, and let n be a positive integer such that

\gcd (q, n) = 1

. The irreducible factors of

x^{n} - 1

and

x^{n} - λ

are fundamental concepts with wide applications in cyclic codes and constacyclic codes. Furthermore, the number of irreducible factors of

x^{n} - 1

and

x^{n} - λ

is useful in many computational problems involving cyclic codes and constacyclic codes. In this paper, we give a more concrete irreducible factorization of

x^{n} - 1

and

x^{n} - λ

. Based on this, the number of irreducible factors of

x^{n} - 1

and

x^{n} - λ

over

F_{q}

, for any

λ \in F_{q}^{*}

, is determined through research on the representatives and the sizes of the q-cyclotomic cosets. As applications, we present the necessary and sufficient conditions for

∣ F (x^{n} - 1) ∣ = 6

and a more concrete factorization of

x^{n} - 1

in these cases.

Keywords: finite fields; irreducible factors; cyclotomic polynomial; cyclotomic cosets; constacyclic codes

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MDPI and ACS Style

Liu, J.; Wu, H. A Note on Factorization and the Number of Irreducible Factors of xⁿ − λ over Finite Fields. Mathematics 2025, 13, 473. https://doi.org/10.3390/math13030473

AMA Style

Liu J, Wu H. A Note on Factorization and the Number of Irreducible Factors of xⁿ − λ over Finite Fields. Mathematics. 2025; 13(3):473. https://doi.org/10.3390/math13030473

Chicago/Turabian Style

Liu, Jinle, and Hongfeng Wu. 2025. "A Note on Factorization and the Number of Irreducible Factors of xⁿ − λ over Finite Fields" Mathematics 13, no. 3: 473. https://doi.org/10.3390/math13030473

APA Style

Liu, J., & Wu, H. (2025). A Note on Factorization and the Number of Irreducible Factors of xⁿ − λ over Finite Fields. Mathematics, 13(3), 473. https://doi.org/10.3390/math13030473

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A Note on Factorization and the Number of Irreducible Factors of xⁿ − λ over Finite Fields

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