Norm residue symbol

Summary

Let $K$ be a local field that contains the group $\mu_n$ of $n$-th roots of unity. The norm residue symbol of degree $n$ is a pairing $(a, b)_n$ of two nonzero elements of $K$ with values in $\mu_n$. It is defined with the reciprocity map of local class field theory, and $(a, b)_n = 1$ exactly when $a$ is a norm from $K(\sqrt[n]{b})$. For $n = 2$ it is the Hilbert symbol. It is a Steinberg symbol, so it factors through $K_2(K)$, and it is the local ingredient of the general reciprocity laws.