Numerous direct calculations were done, and the proof of the Mordell–Weil theorem had to proceed by some surrogate of a finiteness proof for a particular H1 group. The 'twisted' nature of objects over fields that are not algebraically closed, which are not isomorphic but become so over the algebraic closure, was also known in many cases linked to other algebraic groups (such as quadratic forms, simple algebras, Severi–Brauer varieties), in the 1930s, before the general theory arrived. The needs of number theory were in particular expressed by the requirement to have control of a local-global principle for Galois cohomology. This was formulated by means of results in class field theory, such as Hasse's norm theorem. In the case of elliptic curves, it led to the key definition of the Tate–Shafarevich group in the Selmer group, which is the obstruction to the success of a local-global principle. Despite its great importance, for example in the Birch and Swinnerton-Dyer conjecture, it proved very difficult to get any control of it, until results of Karl Rubin gave a way to show in some cases it was finite (a result generally believed, since its conjectural order was predicted by an L-function formula). The other major development of the theory, also involving John Tate was the Tate–Poitou duality result. Technically speaking, G may be a profinite group, in which case the definitions need to be adjusted to allow only continuous cochains. References , translation of Cohomologie Galoisienne, Springer-Verlag Lecture Notes 5 (1964). Algebraic number theory Class field theory Cohomology theories Galois theory Homological algebra