GALOIS COHOMOLOGY

Galois cohomology in its current form took shape during the 1950s as a way of formulating class field theory in a more topological way. This viewpoint was developed by the likes of John Tate, Emil Artin and Gerhard Hochschild. At the 1962 International Congress of Mathematics, (the conference where the Field's medals are awarded) Tate announced several duality theorems about the group cohomology of finite modules and abelian varieties over local and global fields. The famous duality theorem for finite modules (known as local Tate-duality) relies heavily on the work of Nakayama ([11]) while several other results are jointly attributed to Poitou ([10] pg.v). Surprisingly, many of these results were never formally published. Indeed James Milne said the motivation for the writing [10] was to record proofs of these results, while other proofs were given in [6] as reproductions of personal correspondence between Serge Lang and Tate. Despite its modern formalism, results best described using Galois cohomology were already being employed as early as the l890s. Two famous examples include the 90th theorem of Hilbert's famous Zahlbericht which relies heavily on the first cohomology group being trivial. Another such example occurs when k is a field of characteristic p, where p is a nonzero prime, and K is a Galois extension of k. Kummer described the homomorphisms from Gal(Kk) to