Pfister forms in the algebraic and geometric theory of quadratic forms

Nearly 60 years ago, Pfister defined what are now called Pfister forms in quadratic form theory. In addition to their remarkable intrinsic properties, Pfister forms are related to symbols in Galois cohomology and K-theory modulo 2, and are at the heart of the Milnor conjecture. In this paper, we intend to show their importance by means of three explicit examples. They also illustrate the evolution of quadratic form theory from its algebraic aspects to geometry of quadrics—more generally of varieties of isotropic subspaces of a given dimension—and their Grothendieck–Chow motives.