Name or references for minimal $N$ such that $\left(\frac{a}{b}\right)_n = \left(\frac{a}{b'}\right)_n$ whenever $b \equiv b' \bmod (N)$

Question

Let $\left( \dfrac{a}{b} \right)_n$ denote the nth power residue symbol, a generalization of the Legendre symbol. I have recently seen it quoted that there is a minimal ideal $N$ (minimal by ideal inclusion) such that if $b \equiv b' \bmod (N)$, then $\left( \dfrac{a}{b} \right)_n = \left( \dfrac{a}{b'} \right)_n$. This comes up when computing with automorphic forms on metaplectic covers of $GL(2)$.

Further, within the field $\mathbb{Q}(i)$ with ring of integers $\mathbb{Z}[i]$, I've heard it quoted that $N = \left( (1 + i)^3 \right)$.

I was hoping for references concerning a name for this $N$, a proof of its existence in general, and a method for finding $N$ given a reasonable number field. Can you help me out?

Answer 1

This integer is called the conductor of the power residue symbol. It coincides with the conductor of the Kummer extension $K(\sqrt[n]{a})/K$, where $K = {\mathbb Q}(\zeta_n)$ is the field of $n$-th roots of unity. This can be found in all decent books on class field theory, e.g. in Artin-Tate. For methods of computing conductors see the books by Cohen and Gras.