A-9 Roots of Unity and Cyclotomic Polynomial over Rings

In §A-7 and §A-8, we learned about the definition and properties of the $\mu$-th roots of unity and the $\mu$-th cyclotomic polynomial over complex numbers (i.e., $X\in ℂ$) as follows:

• The $𝛍$-th roots of unity are the solutions for $X^{\mu }=1$ over $X\in ℂ$ (complex numbers). The formula for the $\mu$-th root of unity is $X=e^{2\mathit{𝜋𝑖𝑘}∕\mu }$ for all integer $k$ such that $0\le k\le \mu -1$.
• The primitive $𝛍$-th roots of unity (denoted as $𝛚$) are those $\mu$-th roots of unity whose order (§A-4.1) is $\mu$ (i.e., ${\omega }^{\mu }=1$ and ${\omega }^{\frac{\mu }{2}}\ne 1$).
• Given any primitive $\mu$-th roots of unity $\omega$, it can generate all primitive $\mu$-th roots of unity by computing ${\omega }^{k^{\prime }}$ such that $k^{\prime }$ is an integer $0<k^{\prime }<\mu$ and $\mathsf{\text{𝗀𝖼𝖽}}(k^{\prime },\mu )=1$ (Theorem A-7.2.4 in §A-4.2).

• The $𝛍$-th cyclotomic polynomial is defined as a polynomial whose roots are the primitive $\mu$-th roots of unity. That is,

  $${\mathrm{\Phi }}_{\mu }(x)=\prod _{\omega \in P(\mu )}(x-\omega )=\prod _{\begin{array}{c}0\le k\le \mu -1,\\ \text{gcd}(k,\mu )=1\end{array}}(x-{\omega }^k)$$

In this section, we will explain the $\mu$-th cyclotomic polynomial over $X\in {\mathbb{Z}}_p$ (ring), which is structured as follows:

Note that in the $\mu$-th cyclotomic polynomial in both cases of over $X\in ℂ$ and over $X\in {\mathbb{Z}}_p$, each of their roots $\omega$ (i.e., the primitive $\mu$-th root of unity) has the order $\mu$ (i.e., ${\omega }^{\mu }=1$ over $X\in ℂ$, and ${\omega }^{\mu }\equiv 1modp$ over $X\in {\mathbb{Z}}_p$). Also note that each root $\omega$ can generate all roots of the $\mu$-th cyclotomic polynomial by computing ${\omega }^{k^{\prime }}$ such that $\mathsf{\text{𝗀𝖼𝖽}}(k^{\prime },\mu )=1$.

Table 2 compares the properties of the roots of unity and the $\mu$-th cyclotomic polynomial over $X\in ℂ$ (complex numbers) and over $X\in {\mathbb{Z}}_p$ (ring).