The $n$’th cyclotomic polynomial $\Phi_n(x)$ is formed by collecting all the linear factors $x-\zeta$ such that $\zeta$ is the $n$’th primitive root of unity, so we can explicitly define it via 为了进一步调试，你可以提供更多关于你的网站的构建环境和部署方式的信息。此外，确认网站的构建日志中是否有任何错误信息，这些信息通常能提供为什么Markdown文件没有被正确处理的线索。如果使用Jekyll等工具，确保所有必需的插件都已安装，并且_config.yml文件中的配置是正确的。如果使用数学表达式，确认是否已启用MathJax或相似的库来渲染这些表达式为了进一步调试，你可以提供更多关于你的网站的构建环境和部署方式的信息。此外，确认网站的构建日志中是否有任何错误信息，这些信息通常能提供为什么Markdown文件没有被正确处理的线索。如果使用Jekyll等工具，确保所有必需的插件都已安装，并且_config.yml文件中的配置是正确的。如果使用数学表达式，确认是否已启用MathJax或相似的库来渲染这些表达式为了进一步调试，你可以提供更多关于你的网站的构建环境和部署方式的信息。此外，确认网站的构建日志中是否有任何错误信息，这些信息通常能提供为什么Markdown文件没有被正确处理的线索。如果使用Jekyll等工具，确保所有必需的插件都已安装，并且_config.yml文件中的配置是正确的。如果使用数学表达式，确认是否已启用MathJax或相似的库来渲染这些表达式

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Because every $n$’th root of unity is always a $d$’th root of unity for some $d\vert n$, so we can always factor $x^n-1$ into cyclotomic polynomials:

From (2), one immediately concludes that $\Phi_1(x)=x-1$ and for all prime $p$, its corresponding cyclotomic polynomial $\Phi_p(x)$ satisfies

Although we can find the expansions of all cyclotomic polynomials by using (2) repeatedly, this method is very inefficient. In this article, we use some tricks from number theory to reduce the difficulty of the computation of $\Phi_n(x)$ and eventually show that in a wide class of $n$, the expansion of $\Phi_n(x)$ only contains terms of the form $\pm x^k$.

Application of Möbius inversion

Let $\mu(n)$ be the Möbius function. Then from the formula

we can convert $\Phi_n(x)$ into

Observe that $e^{2\pi ir/(n/d)}$ iterates through all the $n/d$’th roots of unity when $r=1,2,\dots,n/d$, so we deduce that

Because the Möbius function is supported on squarefree integers (i.e. not divisible by any square of a prime), we can effectively restrict our focus to cyclotomic polynomials $\Phi_n(x)$ with squarefree $n$. Specifically, there is

Theorem 1: If $n=p_1^{k_1}p_2^{k_2}\cdots p_s^{k_s}$ and $m=p_1p_2\cdots p_s$, then

Reduction to odd squarefree $n$

If $n$ is an even squarefree number, write $m=n/2$, so we can break $d\vert n$ into two disjoint cases:

1. $d$ is even, so $d=2\delta$ for some $\delta\vert m$.
2. $d$ is odd, so $d\vert m$.

Thus, (5) becomes

Because of $x^{m/d}+1=(-1)[(-x)^{m/d}-1]$ and (4), we deduce that

Theorem 2: If $m>1$ is odd, then

Thus, to compute the expansions of cyclotomic polynomial $\Phi_n(x)$, we only need to focus on the situations when $n$ is a product of distinct odd primes, and other cases can be treated via (6) and (7).

The case where $n=pq$

Equation (3) accounts for the case when $n$ is a prime. In this section, we prove that when $n=pq$ for odd primes $p$ and $q$, the expansion of $\Phi_{pq}(x)$ only consists of terms of the form $\pm x^k$.

Plugging $n=pq$ and (3) into (2) gives

so we have

Multiplying both sides by $x^{pq}-1$ and using (3) two more times, we arrive at the formula

Patterns in the coefficients

Because the Diophantine equation $\mu q+\nu p=k$ has at most one solution $(\mu,\nu)$ satisfying $0\le\mu<p$ and $0\le\nu<q$, the expansion of $\Phi_p(x^q)\Phi_q(x^p)$ only contains terms of the form $x^k$. Multiplying by $x-1$ indicates that the right-hand side only contains terms of the form $\pm x^k$.

Since $\deg\Phi_{pq}=(p-1)(q-1)<pq$, the terms of $x^{pq}\Phi_{pq}(x)$ and $-\Phi_{pq}(x)$ do not merge. Collecting all the terms of degree $\le(p-1)(q-1)$ on the right-hand side produces the expansion of $-\Phi_{pq}(x)$. Therefore, we conclude that $\Phi_{pq}(x)$ only contains terms of the form $\pm x^k$.

Combining this with Theorem 1 and 2, we deduce our main result:

Theorem 3 (Migotti 1883¹): Let $n$ be divisible by at most two different odd primes. Then the expansion of $\Phi_n(x)$ only contains terms of the form $\pm x^k$.

Conclusion

In this article, we apply number-theoretic methods and demonstrate that knowing the expansions of $\Phi_n(x)$ for squarefree odd $n$ is sufficient to compute all other cyclotomic polynomials (Theorem 1 and 2). Finally, by investigating the situation when $n$ is a product of two distinct odd primes, we acquire a sufficient condition for cyclotomic polynomials to only possess terms of the form $\pm x^k$ in their expansions (Theorem 3).

Theorem 3 is equivalent to saying that if $\Phi_n(x)$ were to contain terms other than $\pm x^k$, $n$ must be divisible by at least 3 distinct odd primes. Indeed, $105=3\times5\times7$ is such an example:

Since 105 is the smallest integer divisible by 3 distinct odd primes, Theorem 3 also indicates that the expansion of $\Phi_n(x)$ contains terms of the form $\pm x^k$ for all $n\le104$.