at most 2, H1(k,G) = 1. The conjecture is known for a totally imaginary number field (which has cohomological dimension 2). More generally, for any number field k, Martin Kneser, Günter Harder and Vladimir Chernousov (1989) proved the Hasse principle: for a simply connected semisimple group G over k, the map is bijective. Here v runs over all places of k, and kv is the corresponding local field (possibly R or C). Moreover, the pointed set H1(kv,G) is trivial for every nonarchimidean local field kv, and so only the real places of k matter. The analogous result for a global field k of positive characteristic was proved earlier by Harder (1975): for every simply connected semisimple group G over k, H1(k,G) is trivial (since k has no real places). In the slightly different case of an adjoint group G over a number field k, the Hasse principle holds in a weaker form: the natural map is injective. For G = PGL(n), this amounts to the Albert–Brauer–Hasse–Noether theorem, saying that a central simple algebra over a number field is determined by its local invariants. Building on the Hasse principle, the classification of semisimple groups over number fields is well understood. For example, there are exactly three Q-forms of the exceptional group E8, corresponding to the three real forms of E8. See also The groups of Lie type are the finite simple groups constructed from simple algebraic groups over finite fields. Generalized flag variety, Bruhat decomposition, Schubert variety, Schubert calculus Schur algebra, Deligne–Lusztig theory Real form (Lie theory) Weil's conjecture on Tamagawa numbers Langlands classification, Langlands dual group, Langlands program, geometric Langlands program Special group, essential dimension Geometric invariant theory, Luna's slice theorem, Haboush's theorem Radical of an algebraic group Notes References Revised and annotated edition of the 1970 original. Revised and annotated edition of the 1970 original. External links Revised and annotated edition of the 1970 original. Linear algebraic groups Lie groups