A Note on Factorization and the Number of Irreducible Factors of <i>x<sup>n</sup></i> − <i>λ</i> over Finite Fields

A Note on Factorization and the Number of Irreducible Factors of <i>x<sup>n</sup></i> − <i>λ</i> over Finite Fields

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">F</mi><mi>q</mi></msub></semantics></math></inline-formula> be...

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Bibliographic Details
Main Authors:	Jinle Liu, Hongfeng Wu
Format:	Article
Language:	English
Published:	MDPI AG 2025-01-01
Series:	Mathematics
Subjects:	finite fields irreducible factors cyclotomic polynomial cyclotomic cosets constacyclic codes
Online Access:	https://www.mdpi.com/2227-7390/13/3/473
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Summary:	Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">F</mi><mi>q</mi></msub></semantics></math></inline-formula> be a finite field, and let <i>n</i> be a positive integer such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>gcd</mi><mo>(</mo><mi>q</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The irreducible factors of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mi>λ</mi></mrow></semantics></math></inline-formula> are fundamental concepts with wide applications in cyclic codes and constacyclic codes. Furthermore, the number of irreducible factors of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mi>λ</mi></mrow></semantics></math></inline-formula> is useful in many computational problems involving cyclic codes and constacyclic codes. In this paper, we give a more concrete irreducible factorization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mi>λ</mi></mrow></semantics></math></inline-formula>. Based on this, the number of irreducible factors of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mi>λ</mi></mrow></semantics></math></inline-formula> over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">F</mi><mi>q</mi></msub></semantics></math></inline-formula>, for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>∈</mo><msubsup><mi mathvariant="double-struck">F</mi><mi>q</mi><mo>∗</mo></msubsup></mrow></semantics></math></inline-formula>, is determined through research on the representatives and the sizes of the <i>q</i>-cyclotomic cosets. As applications, we present the necessary and sufficient conditions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∣</mo><mi mathvariant="fraktur">F</mi><mo>(</mo><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>∣</mo><mo>=</mo><mn>6</mn></mrow></semantics></math></inline-formula> and a more concrete factorization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> in these cases.
ISSN:	2227-7390