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SEMI-CYCLOTOMIC POLYNOMIALS

LEE, KI-SUK;LEE, JI-EUN;Kim, JI-HYE

  • Received : 2015.09.05
  • Accepted : 2015.11.14
  • Published : 2015.12.25

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

n-th cyclotomic polynomial;semi-cyclotomic polynomial;irreducible polynomial

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