标签：varepsilon Phi right frac Cyclotomic Polynomial pi left
分圆多项式(Cyclotomic Polynomial)
对于任意正整数\(n\)，\(\Phi_n(x)\)是一个不可约的首一多项式，其中\(\Phi_n(x)\)表示第\(n\)个分圆多项式，满足\(\Phi_n(x)│x^n-1\)，任意\(k<n\)，\(\Phi_n(x)∤x^k-1\)。且这个多项式的根都是单位根\(e^{2iπ\frac{k}{n}}\)，所以这个多项式可以写为：
${\Phi_{n}(x)} = {\prod\limits_{\substack{1 \leq k \leq n \\ {\gcd{({k,n})}} = 1}}\left( {x - e^{2i\pi\frac{k}{n}}} \right)}$
例如：
\(\Phi_1(x)=x-1\)
\(\Phi_2(x)=x+1\)
\(\Phi_3(x)=x^2+x+1\)
\(\Phi_4(x)=x^2+1\)
其具有重要性质：
$\Phi_{2^n}(x)=x^{2^{n-1}}+1$
证明：
显然，对于\(k=2ε\)有\(\mbox{gcd}(k,n)≠1\)，而对于\(k=2ε+1\)有\(\mbox{gcd}(k,n)=1\)成立，其中\(ε∈\mathbb{F}_2^{n-1}\)。故有
${\Phi_{2^{n}}(x)} = {\prod\limits_{\substack{1 \leq k \leq 2^{n} \\ {\gcd{({k,2^{n}})}} = 1}}\left( {x - e^{2i\pi\frac{k}{2^{n}}}} \right)} = {\prod\limits_{\varepsilon \in \mathbb{F}_{2}^{n - 1}}\left( {x - e^{2i\pi\frac{2\varepsilon + 1}{2^{n}}}} \right)}$
考虑任意\(b-a=\frac{1}{2}\)，有
$
\begin{matrix}
& {\left( {x - e^{a2i\pi}} \right)\left( {x - e^{b2i\pi}} \right)} \\
= & {\left( {x - e^{a2i\pi}} \right)\left( {x - e^{{({\frac{1}{2} + a})}2i\pi}} \right)} \\
= & {\left( {x - e^{a2i\pi}} \right)\left( {x + e^{a2i\pi}} \right)} \\
= & {x^{2} - e^{2a2i\pi}}
\end{matrix}$
从而
$
{\Phi_{2^{n}}(x)} = {\prod\limits_{\varepsilon \in \mathbb{F}_{2}^{n - 1}}\left( {x - e^{2i\pi\frac{2\varepsilon + 1}{2^{n}}}} \right)} = {\prod\limits_{\varepsilon \in \mathbb{F}_{2}^{n - 2}}\left( {x^{2} - e^{2i\pi\frac{2\varepsilon + 1}{2^{n - 1}}}} \right)} = \cdots = x^{2^{n - 1}} + 1$
证毕。
参考
分圆多项式 cyclotomic polynomial-CSDN博客
分圆多项式（cyclotomic polynomial） - PamShao - 博客园 (cnblogs.com)
标签：varepsilon,
Phi,
right,
frac,
Cyclotomic,
Polynomial,
pi,
left
From： https://www.cnblogs.com/miro-cnblogs/p/18470155