Quadratic forms over nonformally real fields with a finite number of quaternion algebras

Craig McCormack Cordes

Vol. 63 (1976), No. 2, 357–365

DOI: 10.2140/pjm.1976.63.357

Abstract

This paper is concerned with quadratic forms over a nonformally real field, F, of characteristic not two, which has only a finite number, m, of quaternion algebras. The number q = |Ḟ∕Ḟ²| is always assumed to be finite. Of central importance will be the radical, R, which can be defined by R = {a ∈Ḟ∣G(1,−)a = Ḟ} where Ḟ = F −{0} and G(1,−a) is the set of nonzero elements represented by the form x² − ay² over F. The main result here is that a finite Witt ring (and hence the complete quadratic form structure) is determined when m = 2 by the Witt group and the order of Ḟ∕R.

Mathematical Subject Classification

Primary: 10C05, 10C05

Milestones

Received: 23 October 1975

Revised: 18 February 1976

Published: 1 April 1976