@article{Xie2024, 
author = {Weitao Xie and Jiayu Zhang and Wei Cao},
title = {On the number of the irreducible factors of  xn−1 over finite fields},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {9},
pages = {23468-23488},
keywords = {finite fields, normal basis, irreducible factors, Diophantine equation, cyclotomic polynomial},
url = {https://www.sciopen.com/article/10.3934/math.20241141},
doi = {10.3934/math.20241141},
abstract = {Let  Fq be the finite field of  q elements, and  Fqn its extension of degree  n. A normal basis of  Fqn over  Fq is a basis of the form  {α,αq,⋯,αqn−1}. Some problems on normal bases can be finally reduced to the determination of the irreducible factors of the polynomial  xn−1 in  Fq, while the latter is closely related to the cyclotomic polynomials. Denote by  F(xn−1) the set of all distinct monic irreducible factors of  xn−1 in  Fq. The criteria for   |F(xn−1)|≤2have been studied in the literature. In this paper, we provide the sufficient and necessary conditions for   |F(xn−1)|=s,where  s is a positive integer by using the properties of cyclotomic polynomials and results from the Diophantine equations. As an application, we obtain the sufficient and necessary conditions for   |F(xn−1)|=3,4,5.}
}