A-9 Roots of Unity and Cyclotomic Polynomial over Ring

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 $\mathbf{\text{𝛍}}$-th roots of unity are the solutions in $ℂ$ to the equation $X^{\mu }=\; 1$. In other words, all $\mu$-th roots of unity can be written as $X\; =e^{2\pi ik∕\mu }$ for integers $k$ with .
• The primitive $\mathbf{\text{𝛍}}$-th roots of unity (denoted as $\mathbf{\text{𝛚}}$) are those $\mu$-th roots of unity whose order (§A-4.1) is $\mu$ (i.e., ${\omega }^{\mu }=\; 1$ and ${\omega }^d\ne 1$ for any with $d\mid \mu$).
• Given any primitive $\mu$-th roots of unity $\omega$, one can generate all primitive $\mu$-th roots of unity by computing ${\omega }^{k^{\prime }}$ such that $k^{\prime }$ is an integer and $\mathsf{\text{𝗀𝖼𝖽}}(k^{\prime },\mu )\; =\; 1$ (Theorem A-7.2.4 in §A-4.2).

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

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

In this section, we will explain the $\mu$-th cyclotomic polynomial over ${\mathbb{Z}}_p$ (with $p$ prime), 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 $ℂ$ (the complex numbers) and over ${\mathbb{Z}}_p$ (the ring).