Galois symbol

Remarks

The map is well defined: the cup product is multilinear, the target is killed by $m$, and Tate showed that $\delta(a) \cup \delta(1 - a) = 0$ for $a \ne 0, 1$. Concretely, for $\sigma \in \Gal(F^s/F)$ and $\alpha^m = a$, the class $\delta(a)$ is represented by the cocycle $\sigma \mapsto \sigma(\alpha)/\alpha$.

Example: in degree $1$, $h^1_{F, m} \colon F^\times / (F^\times)^m \to H^1(F, \mu_m)$ is an isomorphism, by Kummer theory and Hilbert's Theorem 90. Example: $F = \R$ and $m = 2$. Then $H^*(\R, \Z/2\Z)$ is a polynomial ring $\F_2[x]$ with $x = \delta(-1)$, so $h^n_{\R, 2}(\set{-1, \ldots, -1}) = x^n \ne 0$.

Non-example: let $F$ have characteristic $p > 0$ and let $m = p$. Then the map $x \mapsto x^p$ is injective on $(F^s)^\times$, and $\mu_p$ is trivial. So the Kummer sequence is not exact on the right in general, and the construction does not apply. This is why $m$ must be invertible in $F$.