The Elementary Type Conjecture in Quadratic Form Theory

The systematic study of quadratic forms over an arbitrary field of characteristic = 2 was initiated by Witt [W]. Two milestone papers in the area are a paper of Pfister [P0] relating quadratic forms and orderings and the paper of Milnor [M] pointing out a possible relationship between the Witt ring of quadratic forms and the Galois cohomology of the field (a relationship eventually verified by the work of Voevodsky; see [AE] [P1]). Standard textbooks on the subject include [L] [Sch1] [Sch2]. It was noted early on that different fields can be ‘quadratically equivalent’ in the sense that their quadratic form theory is ‘the same’, e.g., see [H] [C0]. For fixed n ≥ 0, there are only a finite number of quadratically inequivalent fields F with |F ∗/F ∗2| = 2. The present paper is a survey of work related to the so-called ‘elementary type conjecture’ [M2]. If true, this conjecture provides a complete classification of fields with |F ∗/F ∗2| = 2 up to quadratic equivalence. Partial results that have been obtained concerning the elementary type conjecture typically involve arguments of a combinatorial nature. At the same time, recent work on the conjecture has featured Galois–theoretic methods and Galois– theoretic formulations of the conjecture. These connections to Galois theory are described in the final section of the paper.