This paper is an updated summary on old and new results about the cohomology set \(H^1(F, G)\) for a linear algebraic group \(G\) over a field \(F\). In the first part fields of cohomological dimension \(\leq 2\) are considered. The main results are:
Theorem of Steinberg = ex-conjecture I of Serre’s book: \(\mathrm{cd}(F)\leq 1\Rightarrow H^1 (F, G)=0\) for every connected group \(G\).
Conjecture II: \(F\) perfect, \(\mathrm{cd}(F)\leq 2\Rightarrow H^1 (F, G)=0\) for every semisimple simply connected group \(G\).
This conjecture has been proved for local fields \(F\) by Kneser, for global fields by Kneser, Harder and Chernousov (type \(E_8\)). Furthermore it is true for any \(F\) as above if \(G\) is not of type \(E_{6,7,8}\) or a trialitarian \(D_4\). This follows from results of Merkurjev-Suslin and Bayer-Parimala.
The second part is concerned with the existence and construction of certain cohomological invariants, i.e. functorial maps \(H^1 (F, G)\to H^i (F,C)\) where \(i\geq 2\) and \(C\) is a commutative \(\Gamma_F\)- torsion-module. Main examples:
For a non-degenerate quadratic form \(q\) the Hasse-Witt invariant \(w_2\) resp. the Arason invariant \(a: I^3/ I^4\to H^3 (F, \mathbb Z/ 2\mathbb Z)\) induce maps \[ w_2: H^1 (F, O(q))\to H^2 (F, \mathbb Z/ 2\mathbb Z) \quad \text{resp.} \quad a= H^1 (F, \mathrm{Spin}(q))\to H^3 (F, \mathbb Z/ 2\mathbb Z). \] For a central simple algebra \(D\) of rank \(n^2\) over \(F\) Merkurjev-Suslin construct a map \[ m: H^1 (F, \mathrm{SL}_D)= F^*/\mathrm{Nrd}(D^*)\to H^3 (F, \mu_n^{\circledast}) \] (provided the characteristic of \(F\) does not divide \(n\)).
Rost constructs a functorial map \[ r: H^1(F, G)\to H^3 (F, \mathbb Q/ \mathbb Z(2)) \] which can be worked out explicitly for \(G\) of type \(G_2\) and \(F_4\).
In all these cases the invariants yield immediate applications to the classification of the corresponding algebraic objects (having automorphism group \(G\)) over \(F\), i.e. quadratic forms, octonion algebras, exceptional Jordan algebras.
Proofs are omitted or only sketched, but for all statements precise references are given (40 items).
For the entire collection see [Zbl 0811.00012].