Vol. 69, No. 1, 1977

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Spaces of similarities. IV. (s,t)-families

Daniel Byron Shapiro

Vol. 69 (1977), No. 1, 223–244

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.

Mathematical Subject Classification
Primary: 10C05, 10C05
Received: 18 March 1976
Published: 1 March 1977
Daniel Byron Shapiro