Abstract
The n-th cyclotomic polynomial <TEX>${\Phi}_n(x)$</TEX> is irreducible over <TEX>$\mathbb{Q}$</TEX> and has integer coefficients. The degree of <TEX>${\Phi}_n(x)$</TEX> is <TEX>${\varphi}(n)$</TEX>, where <TEX>${\varphi}(n)$</TEX> is the Euler Phi-function. In this paper, we define Semi-Cyclotomic Polynomial <TEX>$J_n(x)$</TEX>. <TEX>$J_n(x)$</TEX> is also irreducible over <TEX>$\mathbb{Q}$</TEX> and has integer coefficients. But the degree of <TEX>$J_n(x)$</TEX> is <TEX>$\frac{{\varphi}(n)}{2}$</TEX>. Galois Theory will be used to prove the above properties of <TEX>$J_n(x)$</TEX>.
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