Galois 1-cocycle
= Galois 1-cocycle
{c}
{title2=$z_{\sigma\tau}=z_\sigma+\sigma z_\tau$}
A continuous one-cocycle for a Galois group acting on an abelian module $M$ is a map satisfying the displayed identity. Cocycles modulo maps $\sigma\mapsto\sigma u-u$ form $H^1(K,M)$ in <Galois cohomology>. For an <elliptic curve>, choosing $mQ=P$ gives the cocycle $\sigma\mapsto\sigma Q-Q$ in $E[m]$, which represents the <Kummer map of an elliptic curve>.