A note on simple decomposition of quadratic forms over linked fields
A field \(F\) is said to be \(n\)-linked if every pair of \(n\)-fold Pfister forms over \(F\) are linked, that is, if every pair of \(n\)-fold Pfister forms have a common factor which is an \((n - 1)\)-fold Pfister form. \textit{R. Elman} and \textit{T. Y. Lam} [Invent. Math. 21, 125-137 (1973; Zbl 0267.10029)] proved that \(F\) is linked if and only if every anisotropic \(\varphi \in I^nF\) has a simple decomposition, that is, if \(\varphi\) can be written as \(\varphi \cong \pi_1 \perp \cdots \perp \pi_r\), where \(\pi_i\)'s are similar to \(n\)-fold Pfister forms (and \(\cong\) is the isometry relation). The present author strengthens this result by showing that \(F\) is \(n\)- linked if and only if every anisotropic \(\varphi \in I^nF\) can be written as the product \(\varphi \cong \pi \otimes \tau\), where \(\pi\) is an \((n - 1)\)-fold Pfister form and \(\tau\) is an even-dimensional quadratic form. As a corollary the author shows that for an \(n\)-linked field \(F\) and its extension field \(K\) the Witt index over \(K\) of every anisotropic form \(\varphi \in I^nF\) is a multiple of \(2^{n - 1}\) (including zero).
scientific article; zbMATH DE number 783945