Abstract |
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Let F be a field of characteristic not 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 ∈ I^3F 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 ∈ 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; 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.
In this article, we investigate how these properties D(n) relate to each other and we show how one can construct fields which fail to have property D(n), n > 2, by starting with a field which fails to have property D(2) and then passing to transcendental field extensions. Particular emphasis is devoted to the situation where K is a field with a discrete valuation with residue field k of characteristic not 2. Here, we study how the properties D(n) behave when one passes from K to k or vice versa. We conclude with some applications and an explicit and detailed example involving rational function fields of transcendence degree at most four over the rationals. |