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.



Computing mth power residue symbols
I am not sure what you mean by "to resort to these symbols". If you want to compute their values given $\alpha$ and ${\mathfrak b}$, just use the definition.


Explicit reciprocity laws for higher powers do exist, but have a couple of natural drawbacks. If you look at the computation of $(\frac ab)$ in integers, you will notice that the process of inverting and reducing essentially boils down to applying the Euclidean algorithm. This also works for $n = 3, 4, 5, 8$ and a few other exponents as well, but not in general. As KConrad explained in his comments, reciprocity laws in the sense of Legendre only apply to principal ideals (or, more generally, to ideals whose order in the class group is coprime to $n$).


The natural way of looking at reciprocity in number fields is the modularity property: you have $(\alpha/{\mathfrak a})\_n = (\alpha/{\mathfrak b})\_n$ if the ideals ${\mathfrak a}$ and ${\mathfrak b}$ lie in the same ray class determined by the abelian extension $K(\sqrt[n]{\alpha})$ of $K = {\mathbb Q}(\zeta\_n)$. I have discussed the case $n = 2$ in detail in my recent book on quadratic number fields.


If you absolutely must use classical reciprocity you should learn about Hilbert symbols. In any case, you cannot avoid class field theory if you're interested in higher reciprocity laws.


