Semi-cyclotomic polynomials. (English) Zbl 1331.11020

Lee, Ki-Suk; Lee, Ji-Eun; Kim, Ji-Hye

Honam Math. J. 37, No. 4, 469-472 (2015).

Summary: The \(n\)-th cyclotomic polynomial \({\Phi}_n(x)\) is irreducible over \(\mathbb{Q}\) and has integer coefficients. The degree of \({\Phi}_n(x)\) is \({\varphi}(n)\), where \({\varphi}(n)\) is the Euler Phi-function. In this paper, we define the semi-cyclotomic polynomial \(J_n(x)\). \(J_n(x)\) is also irreducible over \(\mathbb{Q}\) and has integer coefficients. But the degree of \(J_n(x)\) is \(\frac{{\varphi}(n)}{2}\). Galois theory will be used to prove the above properties of \(J_n(x)\).

MSC:

11C08	Polynomials in number theory
11R09	Polynomials (irreducibility, etc.)
11R32	Galois theory

Keywords:

\(n\)-th cyclotomic polynomial; semi-cyclotomic polynomial; irreducible polynomial