Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

SEMI-CYCLOTOMIC POLYNOMIALS

  • LEE, KI-SUK (Department of Mathematics Education, Korea National University of Education) ;
  • LEE, JI-EUN (Myogok Elementary School) ;
  • Kim, JI-HYE (Department of Mathematics Education, Korea National University of Education)
  • Received : 2015.09.05
  • Accepted : 2015.11.14
  • Published : 2015.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

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 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 $J_n(x)$.

Keywords

References

  1. J. R. Bastida and R. Lyndon, Field Extensions and Galois Thory, Encyclopedia of Mathmatics and Its Application, Addison-Wesley Publishing Company (1984).
  2. T. W. Hungerford, Abstract Algebra An Introduction, Brooks/Cole, Cengage Learning (2014).
  3. S. Lang, Algebra, Addison-Wesley Publishing Company (1984).
  4. P. Ribenboim, Algebraic Numbers, John Wiley and Sons Inc (1972).