Cohomological invariants in Galois cohomology, by Skip Garibaldi, Alexander

Quadratic forms q(x1, . . . , xn) over a field k are the classical examples of nonlinear scalar-valued functions on a vector space V = k, and they lie at the foundation of nineteenth-century analytic geometry. Motivated by problems such as finding the axes of a conic, geometers studied quadratic forms under symmetries of V . Given two quadratic forms q, q′ on V , how may we tell if q and q′ are equivalent over k, that is, if q′ = q ◦ T for T ∈ GLn(k)? The discriminant is a place to start. The discriminant disc(q) of a quadratic form over k (which we always assume is of characteristic not 2) is the square class of the determinant of the symmetric matrix A such that q(x) = xAx, x ∈ V : disc(q) = det(A) mod k∗2 ∈ k∗/k∗2 ∪ {0}. (Here k∗ denotes the multiplicative group of the field k.) Applying the transformation q → q ◦ T induces a transformation A → T AT , and we deduce that disc(q′) = disc(q) from det(T AT ) = det(A) det(T ). Bourbaki observes that it was the visibility of this sort of behavior of the determinant under linear transformations, thanks to Gauss [4, 301–302], that gave “la première impulsion à la théorie générale des invariants” [2, 163–164]. The basic question of invariant theory—going back to the nineteenth century as well—is to determine which polynomial functions f : V m → k are left unchanged by composition with elements of a given group G of linear transformations on V . (See Weyl’s landmark monograph [14].) Given a quadratic form q with symmetric matrix A, the determinant det(A) is a polynomial function of the quadratic form (and of A) unchanged by composition with elements from SLn(k). Then, working modulo squares, the discriminant disc(q) is an invariant unchanged by composition with elements of GLn(k)—an invariant of equivalence classes of quadratic forms. A collection of additional invariants of quadratic forms emerged from what we now consider classical quadratic form theory, from the early to mid-twentieth century. Over certain fields, these invariants are enough to classify nondegenerate quadratic forms up to equivalence. Let q be a nondegenerate quadratic form, that is, with nonzero discriminant. The rank of q is the dimension of its underlying vector space. Due to Sylvester, the signature of q is, for k ⊂ R, the pair (r, s) where r is the dimension of the maximal subspace P on which q|P ≥ 0 and s is the dimension of the maximal subspace N on which q|N ≤ 0; we have that r + s = n. The Witt index of q is the dimension of a maximal subspace on which the quadratic form is zero. For instance, the Witt index of q(x1, x2) = x1x2 is 1.