Modified Cyclotomic Polynomial and Its Irreducibility

Abstract

Finding irreducible polynomials over Q (or over Z ) is not always easy. However, it is well-known that the mth cyclotomic polynomials are irreducible over Q . In this paper, we define the mth modified cyclotomic polynomials and we get more irreducible polynomials over Q 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 mth modified cyclotomic polynomials when m = p α with p a prime number and α a positive integer.

Keywords

• irreducible polynomial
• cyclotomic polynomial
• modified cyclotomic polynomial
• semi-cyclotomic polynomial
• multiplicative group