Infinite-dimensional quadratic forms admitting composition

Abstract

In [2] Albert proved that a finite-dimensional absolute-valued algebra over the reals is necessarily aiternative (and hence the reals, complexes, quaternions, or Cayley numbers). In [3 ] he extended this from finite-dimensional to algebraic algebras. Recently, Wright [9] succeeded in removing the assumption that the algebra is algebraic. Wright proceeds by proving that the norm springs from an inner product, and then that the algebra is algebraic. Now if the norm I xI comes from an inner product, then xI 2 is a quadratic form in x, and moreover the assumption xy = x I y| means that it is a quadratic form admitting composition. Thus Albert's finite-dimensional theorem can be proved by combining Wright's result with Hurwitz's classical theorem on quadratic forms admitting composition. The main purpose of this paper is to make a similar method possible in the infinite-dimensional case by providing a suitable generalization of Hurwitz's theorem. The result is essentially that infinite-dimensional quadratic forms cannot admit composition, except in the rather trivial case of purely inseparable fields of characteristic two. In the concluding moments of the proof we rely heavily on the recent developments in the theory of alternative rings. Until then the paper is quite self-contained and elementary, and the style of the argument is very close to that of [1], [5], and [8]. Let A be a vector space over a field F. The function g(x) from A to F is a quadratic form if