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

Detlef W. Hoffmann ; Jean-Pierre Tignol

Documenta mathematica, Tome 3 (1998), pp. 189-214

Résumé

Let F be a field of characteristic =2. We define certain properties D(n),n∈{2,4,8,14}, of F as follows : F has property D(14) if each quadratic form φ∈I3F of dimension 14 is similar to the difference of the pure parts of two 3-fold Pfister forms; F has property D(8) if each form φ∈I2F 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; F has property D(4) if any two 4-dimensional forms over F of the same determinant which become isometric over some quadratic extension always have (up to similarity) a common binary subform; F has property D(2) if for any two binary forms over F and for any quadratic extension E/F we have that if the two binary forms represent over E a common nonzero element, then they represent over E a common nonzero element in F. Property D(2) has been studied earlier by Leep, Shapiro, Wadsworth and the second author. In particular, fields where D(2) does not hold have been known to exist.

Détail

DOI : 10.4171/dm/40

Classification : 11E04, 11E16, 11E81, 16K20