Let's say I have a two odd primes, $p, q$ and $K$ is the field $\mathbb{Q}(\zeta_{pq})$. Let's say $\alpha \in \mathcal{O}$ is an **arbitrary** element in the ring of integers of $K$, $\frak{b} \subset \mathcal{O}$ is a prime ideal of $\mathcal{O}$, and $\alpha \notin \frak{b}$. Not sure what it's called but I'd like to compute a law that would allow me to "flip" the numerator and denominator in the following residue symbol:

$\Big(\frac{\alpha}{\frak{b}}\Big)_{q} = \alpha^{\frac{N(\frak{b}) - 1}{q}} \text{mod } \frak{b}$.

In my personal studies I've found I've had no other options but to resort to these symbols. I'm rather unfamiliar with any other reciprocity laws aside from quadratic, cubic, and quartic and any references or suggestions to learn based on where I stand would be appreciated.