The determination of spaces
of similarities is a generalization of the Hurwitz problem of compasition of quadratic
forms. For forms σ, q over the field F, we write σ <Sim(q) if q admits composition
with σ. When F is the real or complex field, the possible dimensions of
σ and q were determined long ago by Radon and Hurwitz. We show that
these classical bounds are still correct over any field F of characteristic not
2.
This paper deals with the more delicate question of which quadratic forms σ, q
over F can admit composition. The motivation of much of this work is the Pfister
factor conjecture: if q is a form of dimension 2m, and σ <Sim(q) for some form σ of
large dimension, then q must be a Pfister form. We prove this true in general when
m ≦ 5, and we also prove it true for all m for a certain class of fields which includes
global fields.
