Quadratic forms and their use in geometry and number theory have a long and distinguished history going back to ancient times. The algebraic theory is more recent. It originates from the work of Ernst Witt in the 1930s, who organized the non-degenerate quadratic forms over a field K into what we now call the Witt ring of K. In the 1960s Albrecht Pfister introduced the basic building blocks of this ring, which we now call Pfister forms.

The d-h Pfister number Pf_{d}(q) of a quadratic form q is the smallest number of d-fold Pfister forms required to represent q. (Here I am assuming that q is contained in the d-th power of the fundamental ideal.) This number is an important measure of the complexity of q. In this talk I will define all of the above terms, including “Witt ring”, “Pfister form” and “fundamental ideal”. I will then discuss a recent result, proved jointly with Patrick Brosnan and Angelo Vistoli, which shows that Pf_{3}(q) can be surprisingly large.