\'Etale algebras and the Kummer theory of finite Galois modules

Galois cohomology groups Hⁱ (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 GK, which in many cases of interest (e. g. K = Q) is too large for computation or even for gaining an intuitive grasp. However, for finite M, an element of Hⁱ (K, M) can be described by a finite amount of data. For the important case i = 1, the appropriate object is an \'etale algebra over K (a finite product of separable field extensions) whose Galois group is a subgroup of the semidirect product Hol M = M Aut M (often called the holomorph of M), equipped with a little bit of combinatorial data. Although the correspondence between H¹ 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 \'etale algebra perspective illuminates some common uses of H¹, including parametrizing cubic and quartic algebras as well as computing the Tate pairing on Galois coclasses of local fields.