Editing Hasse principle (section)

==Hasse principle for algebraic groups==

The Hasse principle for [[algebraic group]]s states that if ''G'' is a simply-connected algebraic group defined over the [[global field]] ''k'' then the map
:<math> H^1(k,G)\rightarrow\prod_s H^1(k_s,G)</math>
is injective, where the product is over all places ''s'' of ''k''.

The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms.

{{harvtxt|Kneser|1966}} and several others verified the Hasse principle by case-by-case proofs for each group. The last case was the group [[E8 (mathematics)|''E''<sub>8</sub>]] which was only completed by {{harvtxt|Chernousov|1989}} many years after the other cases.

The Hasse principle for algebraic groups was used in the proofs of the [[Weil conjecture for Tamagawa numbers]] and the [[strong approximation theorem]].