Galois module

A discrete Galois module over $K$ is an abelian group with a continuous action of the absolute Galois group $G_K$. Continuity means that each element has an open stabilizer. Examples are the algebraic points of an elliptic curve, its finite torsion module $E[m]$, and the group of roots of unity ${\mu }_m$. Galois cohomology takes continuous cocycles with values in such a module.

The module ${\mu }_m$ consists of the $m$th roots of unity in a separable closure. If the field characteristic does not divide $m$, it has $m$ elements and fits into the Kummer sequence $1\to {\mu }_m\to K_s^{\times }\stackrel{m}{\to }{K}_s^{\times }\to 1$. Its Galois action is the cyclotomic action; a generator gives a noncanonical identification with $\mathbb{Z}/m\mathbb{Z}$.