Norm residue symbol

Remarks

The value lies in $\mu_n$ because $\theta(a)(\beta)$ is again an $n$-th root of $b$, and it does not depend on the choice of $\beta$, since $\theta(a)$ fixes $\mu_n \subseteq K$. The symbol is multiplicative in each variable, satisfies $(a, 1 - a)_n = 1$ for $a \ne 0, 1$, and $(a, b)_n = 1$ if and only if $a$ is a norm from $K(\beta)$ to $K$. Some books use $\theta(b)$ acting on an $n$-th root of $a$ instead; the two conventions give inverse values.

Example: for $n = 2$ and $K = \R$, the reciprocity map sends $a < 0$ to complex conjugation and $a > 0$ to the identity of $\Gal(\C/\R)$. For $a = b = -1$ and $\beta = i$, $(-1, -1)_2 = \bar{i}/i = -1$, the value of the Hilbert symbol.

Non-example: the field $\Q_3$ does not contain a primitive cube root of unity, since $\mu(\Q_3) = \set{\pm 1}$. So the norm residue symbol of degree $3$ is not defined on $\Q_3$; the definition needs $\mu_n \subseteq K$.