automorphism group is of order 2, switching the two "legs" of the Dn diagram. The action of the absolute Galois group of k on the Dynkin diagram is trivial if and only if the signed discriminant d of q in k*/(k*)2 is trivial. If d is nontrivial, then it is encoded in the Galois action on the Dynkin diagram: the index-2 subgroup of the Galois group that acts as the identity is . The group G is split if and only if q has Witt index n, the maximum possible, and G is quasi-split if and only if q has Witt index at least n − 1. Torsors and the Hasse principle A torsor for an affine group scheme G over a field k means an affine scheme X over k with an action of G such that is isomorphic to with the action of on itself by left translation. A torsor can also be viewed as a principal G-bundle over k with respect to the fppf topology on k, or the étale topology if G is smooth over k. The pointed set of isomorphism classes of G-torsors over k is called H1(k,G), in the language of Galois cohomology. Torsors arise whenever one seeks to classify forms of a given algebraic object Y over a field k, meaning objects X over k which become isomorphic to Y over the algebraic closure of k. Namely, such forms (up to isomorphism) are in one-to-one correspondence with the set H1(k,Aut(Y)). For example, (nondegenerate) quadratic forms of dimension n over k are classified by H1(k,O(n)), and central simple algebras of degree n over k are classified by H1(k,PGL(n)). Also, k-forms of a given algebraic group G (sometimes called "twists" of G) are classified by H1(k,Aut(G)). These problems motivate the systematic study of G-torsors, especially for reductive groups G. When possible, one hopes to classify G-torsors using cohomological invariants, which are invariants taking values in Galois cohomology with abelian coefficient groups M, Ha(k,M). In this direction, Steinberg proved Serre's "Conjecture I": for a connected linear algebraic group G over a perfect field of cohomological dimension at most 1, H1(k,G) = 1. (The case of a finite field was known earlier, as Lang's theorem.) It follows, for example, that every reductive group over a finite field is quasi-split. Serre's Conjecture II predicts that for a simply connected semisimple group G over a field of cohomological dimension