Modified Cyclotomic Polynomial and Its Irreducibility

• Published in: Mathematics
• Main Authors: Ki-Suk Lee, Sung-Mo Yang, Soon-Mo Jung
• Format: Article
• Language: English
• Published: MDPI AG 2020-03-01
• Subjects: irreducible polynomial; cyclotomic polynomial; modified cyclotomic polynomial; semi-cyclotomic polynomial; multiplicative group
• Online Access: https://www.mdpi.com/2227-7390/8/3/343

Finding irreducible polynomials over <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Q</mi> </semantics> </math> </inline-formula> (or over <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Z</mi> </semantics> </math> </inline-formula>) is not always easy. However, it is well-known that the <i>m</i>th cyclotomic polynomials are irreducible over <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Q</mi> </semantics> </math> </inline-formula>. In this paper, we define the <i>m</i>th modified cyclotomic polynomials and we get more irreducible polynomials over <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Q</mi> </semantics> </math> </inline-formula> systematically by using the modified cyclotomic polynomials. Since not all modified cyclotomic polynomials are irreducible, a criterion to decide the irreducibility of those polynomials is studied. Also, we count the number of irreducible <i>m</i>th modified cyclotomic polynomials when <inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>=</mo> <msup> <mi>p</mi> <mi>&#945;</mi> </msup> </mrow> </semantics> </math> </inline-formula> with <i>p</i> a prime number and <inline-formula> <math display="inline"> <semantics> <mi>&#945;</mi> </semantics> </math> </inline-formula> a positive integer.