On 14-dimensional quadratic forms in I3I^3, 8-dimensional forms in I2I^2, and the common value property

  • Detlef W. Hoffmann

  • Jean-Pierre Tignol

    25030 Besancon Cedex, France
On 14-dimensional quadratic forms in $I^3$, 8-dimensional forms in $I^2$, and the common value property cover
Download PDF

This article is published open access.

Abstract

Let FF be a field of characteristic 2\neq 2. We define certain properties D(n),n{2,4,8,14}D(n), n\in\{ 2,4,8,14\}, of FF as follows : FF has property D(14)D(14) if each quadratic form φI3F\varphi\in I^3F of dimension 1414 is similar to the difference of the pure parts of two 3-fold Pfister forms; FF has property D(8)D(8) if each form φI2F\varphi\in I^2F of dimension 8 whose Clifford invariant can be represented by a biquaternion algebra is isometric to the orthogonal sum of two forms similar to 2-fold Pfister forms; FF has property D(4)D(4) if any two 4-dimensional forms over FF of the same determinant which become isometric over some quadratic extension always have (up to similarity) a common binary subform; FF has property D(2)D(2) if for any two binary forms over FF and for any quadratic extension E/FE/F we have that if the two binary forms represent over EE a common nonzero element, then they represent over EE a common nonzero element in FF. Property D(2)D(2) has been studied earlier by Leep, Shapiro, Wadsworth and the second author. In particular, fields where D(2)D(2) does not hold have been known to exist.

Cite this article

Detlef W. Hoffmann, Jean-Pierre Tignol, On 14-dimensional quadratic forms in I3I^3, 8-dimensional forms in I2I^2, and the common value property. Doc. Math. 3 (1998), pp. 189–214

DOI 10.4171/DM/40