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Semi-cyclotomic polynomials. (English) Zbl 1331.11020
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
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