Review of Kneser’s work on algebraic groups and the Hasse principle and subsequent developments

Let k be a number field. Let Ω denote the set of places of k and for v ∈ Ω, let kv denote the completion of k at v. A classical theorem of Hasse-Minkowski states that a quadratic form q over k represents zero non-trivially provided it represents zero non-trivially over kv for all v ∈ Ω; in particular, two quadratic forms over k are isomorphic if they are isomorphic over kv for all v ∈ Ω. Another classical theorem of Hasse-Brauer-Noether states that two central simple algebras over k are isomorphic if they are isomorphic over kv for all v ∈ Ω a consequence of the injectivity of the map Br(k) → ⊕ v∈Ω Br(kv), Br(k) denoting the Brauer group of k. These results can be formulated as a Hasse principle for Galois cohomology.