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

Jinle Liu and Hongfeng Wu ()
Additional contact information
Jinle Liu: College of Science, North China University of Technology, Beijing 100144, China
Hongfeng Wu: College of Science, North China University of Technology, Beijing 100144, China

Mathematics, 2025, vol. 13, issue 3, 1-23

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 ∈ 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/3/473/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/3/473/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:3:p:473-:d:1580965

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().