Let $L = \mathbb{Q}(\omega_q)$ where $\omega_q$ is a primitive $q$th root of unity. Then: 1. $[L:\mathbb{Q}] = \varphi(q)$, where $\varphi(q) = |(\mathbb{Z}/q\mathbb{Z})^\times|$ is Euler's totient. 2. $L/\mathbb{Q}$ is Galois with $\operatorname{Gal}(L/\mathbb{Q}) \cong (\mathbb{Z}/q\mathbb{Z})^\times$, where $r \in (\mathbb{Z}/q\mathbb{Z})^\times$ acts by $\omega_q \mapsto \omega_q^r$. 3. The ring of integers is $\mathcal{O}_L = \mathbb{Z}[\omega_q] = \mathbb{Z}[x]/\Phi_q(x)$, where $\Phi_q(x)$ is the $q$th cyclotomic polynomial. 4. A prime $p$ ramifies in $\mathcal{O}_L$ if and only if $p \mid q$. 5. If $p \nmid q$, then $(p)$ factors as a product of $\varphi(q)/f$ distinct prime ideals, each of norm $p^f$, where $f$ is the order of $p$ in $(\mathbb{Z}/q\mathbb{Z})^\times$.