Étale algebras and the Kummer theory of finite Galois modules

Abstract

Galois cohomology groups $H^i(K,M)$ are widely used in algebraic number theory, in such contexts as Selmer groups of elliptic curves, Brauer groups of fields, class field theory, and Iwasawa theory. The standard construction of these groups involves maps out of the absolute Galois group $G_K$, which in many cases of interest (e.g. $K = \mathbb{Q}$) is too large for computation or even for gaining an intuitive grasp. However, for finite $M$, an element of $H^i(K,M)$ can be described by a finite amount of data. For the important case $i = 1$, the appropriate object is an étale algebra over $K$ (a finite product of separable field extensions) whose Galois group is a subgroup of the semidirect product $\operatorname{Hol} M = M \rtimes \operatorname{Aut} M$ (often called the \emph{holomorph} of $M$), equipped with a little bit of combinatorial data. Although the correspondence between $H^1$ and field extensions is in widespread use, it includes some combinatorial and Galois-theoretic details that seem never to have been written down. In this short quasi-expository paper, we fill in this gap and explain how the étale algebra perspective illuminates some common uses of $H^1$, including parametrizing cubic and quartic algebras as well as computing the Tate pairing on Galois coclasses of local fields.