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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?

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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.

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  • $\begingroup$ Thank you! Now I feel a bit guilty, because I've seen and maybe even said the word "conductor" quite a bit in other contexts, but haven't followed up on it yet. It's time to rectify that! (I really should learn class field theory sometime too) $\endgroup$ – davidlowryduda Dec 7 '13 at 21:40
  • $\begingroup$ I also suggest looking at Roquette's manuscript "What is reciprocity" as soon as it reappears on his site rzuser.uni-heidelberg.de/~ci3/manu.html $\endgroup$ – Franz Lemmermeyer Dec 8 '13 at 20:02

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