On the indecomposable quadratic forms of the image of the homomorphism \(W(F) \to W(F(\varphi))\)
Let \(F\) be a field of characteristic not \(2\), \(\varphi\) a nondegenerate quadratic form over \(F\) and \(F(\varphi)\) the function field of the affine quadric defined by \(\varphi =0\) over \(F\). In this paper, the author studies the image \(r_\varphi(W(F))\) of the restriction homomorphism \(r_\varphi:W(F)\to W(F(\varphi))\) between the Witt rings of \(F\) and of \(F(\varphi)\). In [\(K\)-Theory 29, No. 4, 253--284 (2003; Zbl 1108.11033)], \textit{B. Kahn} and the author studied the following question. Let \(\psi\in r_\varphi(W(F))\). What is the minimal dimension of a form \(\psi'\) over \(F\) such that \(\psi'_{F(\varphi)}\) is Witt equivalent to \(\psi\)\,? In the present paper, the author studies what he calls indecomposable forms in \(r_\varphi(W(F))\), i.e. forms \(\psi\in r_\varphi(W(F))\) such that there do \textit{not} exist forms \(\psi_1,\psi_2\in r_\varphi(W(F))\) of dimension \(>0\) such that \(\psi\cong\psi_1\perp\psi_2\). The set of such indecomposable forms (resp. of such forms of dimension \(n\)) will be denoted by \(\text{Ind}(\varphi)\) (resp. \(\text{Ind}_n(\varphi)\)). Of course, any \(1\)-dimensional form over \(F\) will give rise to an indecomposable form over \(F(\varphi)\). Furthermore, in case \(\text{Ind}_1(\varphi)\subsetneq \text{Ind}(\varphi)\), the author defines the integer \(m(\varphi)=\min\{\dim\psi\,|\,\psi\in \text{Ind}(\varphi),\;\dim\psi>1\}\). Recall that an extension \(L/F\) is called excellent if for any form \(\psi\) over \(F\), the anisotropic part of \(\psi_{L}\) is defined over \(F\). It follows that \(\text{Ind}_1(\varphi)= \text{Ind}(\varphi)\) iff \(F(\varphi)/F\) is excellent. This is always true if \(\dim\varphi =2,3\) or if \(\varphi\) is similar to a \(2\)-fold Pfister form. It never holds if \(\varphi\) is not a Pfister neighbor, in which case it is shown that \(m(\varphi)\leq\dim (\varphi_{F(\varphi)})_{\mathrm{an}}\). But for any \(n\geq 5\), \textit{O. T. Izhboldin} [Doc. Math., J. DMV 1, 127--136 (1996; Zbl 0857.11017)] has given examples of fields \(F\) and of Pfister neighbors \(\varphi\) over \(F\) of dimension \(n\) such that \(F(\varphi)/F\) is not excellent. In this situation, the author gives some upper and lower bounds on \(m(\varphi)\) but the results seem somewhat inconclusive. In the case of non-Pfister neighbors of dimension \(4\leq n\leq 8\), the author computes \(m(\varphi)\) and describes \(\text{Ind}_{m(\varphi)}(\varphi)\) completely for \(4\leq n\leq 7\) and partially for \(n=8\).
scientific article; zbMATH DE number 6108411
scientific article