Cohomological invariants associated to symmetric bilinear forms

This paper is an outgrowth of the notes for a lecture given at the Mathematics Department of Queen’s University in December of 1988. I would like to take this opportunity to thank the members of that department for their hospitality during my visit. I would also like to thank Bruno Kahn for reading early versions of this manuscript and making several helpful remarks. The basic idea of the lecture was to summarize what was known (at least to me) at the time about characteristic classes in the mod 2 Galois cohomology of fields K of characteristic not equal to 2 which arise either from symmetric bilinear forms or representations of Galois groups in their orthogonal groups. Some new results have appeared during the writing of this paper; they are discussed below. The theory, at least through the lens of the current discussion, has a long, though somewhat desultory pedigree. The most prominent classical cohomological invariants are the Delzant Stiefel-Whitney classes [1] of a symmetric bilinear form. More recently, Fröhlich [2] introduced the spinor class Sp2(ρ) associated to an orthogonal representation ρ : G → On(K) of a Galois group G of a finite Galois extension of K. Sp2(ρ) is defined to be the image in the Galois cohomology group H 2 et(K,Z/2) under cup product of the element in H(G,H et(K,Z/2)) defined by the composite