Given $r \in \mathbb{N}$, prove there exist odd primes $p,q$ s.t. $p$ splits into $r$ primes in the $q$th cyclotomic field.
Let $\omega = e^{2\pi i /q}$. Then I know that $(p,q) = 1 \implies p \nmid \text{disc}(\omega)$ so $p$ is unramified, implying that I simply need to assure that $f = \frac{q - 1}{r}$. In cyclotomic fields, $f$ is the multiplicative order of $p$ in $\mathbb{Z}/q\mathbb{Z}^\times$; thus, I need to show that there exist odd primes $p,q$ s.t. $$ p^{\frac{q-1}{r}} \equiv 1 \pmod{q} $$
and that for any $a < \frac{q-1}{r}$, $p^a \not\equiv 1 \pmod{q}$. I'm not sure how to make this happen, or if this even the right path; any help is appreciated.
Solution (based on the comments below): Fix a prime $q$ s.t. $q \equiv 1 \pmod{r}$; we know such a prime exists via Dirichlet's Theorem. Then since $(\mathbb{Z}/q\mathbb{Z})^\times$ is cyclic, there exists an $a \in (\mathbb{Z}/q\mathbb{Z})^\times$ s.t. $|a| = \frac{q-1}{r}$. Now, invoking Dirichlet's Theorem once more, fix a prime $p$ s.t. $p \equiv a \pmod{q}$. Then the order of $p$ in $(\mathbb{Z}/q\mathbb{Z})^\times$ is $\frac{q-1}{r}$ as desired.