Isotropy of quadratic forms over the function field of a quadric
Let \(F\) be a field of characteristic different from 2. This paper deals with the question whether an anistropic quadratic form \(\varphi\) over \(F\) becomes isotropic over the function field \(F(\psi)\) of another quadratic form \(\psi\) over \(F\). The first main theorem says that if there exists \(n\geq 0\) such that \(\dim\varphi\leq 2^n< \dim\psi\) then \(\varphi\) stays anisotropic over \(F(\psi)\). As an immediate consequence one has that if \(\psi\) is anisotropic with \(\dim\psi= 2^n+ m\), where \(n\geq 0\) and \(1\leq m\leq 2^n\) then the Witt index of \(\psi\otimes F(\psi)\) is less than or equal to \(m\). The second main theorem states that if \(\varphi\) is an anisotropic form over \(F\) with \(\dim \varphi\leq 2^n+ 1\) then there exists a field extension \(K/F\) with an anisotropic \((n+ 1)\)-fold Pfister form \(\pi\) over \(K\) such that \(\varphi\otimes K\) is a subform of \(\pi\). In fact, if there exists an anisotropic \((n+ 1)\)-fold Pfister form \(\pi\) already over \(F\) then one can construct \(K/F\) such that \(\pi\otimes K\) stays anisotropic and \(\varphi\otimes K\) is a subform of \(\pi\otimes K\). An example shows that there exists a field \(F\) with an anisotropic form \(\varphi\) with \(\dim\varphi= 2^n+ 2\), \(n\geq 2\), such that there does not exist \(K/F\) with \(\varphi\otimes K\) an anisotropic Pfister neighbor. This shows that generally the above result cannot be improved. The third main theorem states that if \(n\geq 3\) and \(\psi\) is an anisotropic form over \(F\) with \(\dim\psi= 2^n+ 1\), and if \(\gamma\) is a form over \(F\) with \(\dim\gamma\geq 2^{n+ 1}- 3\) then \(\psi\otimes F(\gamma)\) is isotropic if and only if there exists an \((n+ 1)\)-fold Pfister form \(\rho\) over \(F\) such that \(\psi\) and \(\gamma\) are Pfister neighbors of \(\rho\). The crucial ingredient used in the proofs of these results is a lemma which says that if \(\varphi\) is anisotropic over \(F\) with \(\dim\varphi\leq 2^n\) then there exists a field extension \(K/F\) with an anisotropic \((n+ 1)\)-fold Pfister form \(\pi\) over \(K\) such that \(\varphi\otimes K\) is a subform of \(\pi\) and such that all anisotropic forms over \(F\) stay anisotropic over \(K(\pi)\). Using the first theorem an ``intrinsic'' characterization of Pfister neighbors is provided. Furthermore, it is shown that the second and third main theorem also hold for anisotropic forms of dimension \(2^n+ m\), \(1\leq m\leq 2^n\), which attain maximal possible Witt index \(m\) over their own function field.
scientific article; zbMATH DE number 856576