Quadratic forms of height and degree 2
Let \(F\) be a field of characteristic not 2. The height \(h(q)\) of a quadratic form \(q\) over \(F\) is defined by: (i) \(h(q)=h(q_{an})\), where \(q_{an}\) is the anisotropic part of \(q\); (ii) If \(q\) is anisotropic \(h(q)=0\) if and only if \(\dim q\leq 1\); (iii) If \(q\) is anisotropic and of dimension \(\geq 2\), \(h(q)= h(q_{F(q)})+1\) (with \(F(q)\) the function field of the quadratic associated to \(q\)). The degree \(\deg(q)\) satisfies by definition: (i) \(\deg(q)= \deg(q_{an})\); (ii) \(\deg(q)=0\) if \(\dim q\) is odd; (iii) If \(q\) is anisotropic \(\dim q\) is even and if \(h(q)=1\) then \(\dim q=2^\delta\). If this is the case \(\deg q=\delta\); (iv) If \(q\) is anisotropic and \(h(q)>1\), \(\deg(q)= \deg(q_{F(q)})\). The main theorem of the paper classifies all forms of height 2 and degree 2, thereby completing earlier results of Knebusch and Fitzgerald. It is shown that an anisotropic form \(q\) with \(h(q)= \deg(q)=2\) is either excellent or a 6-dimensional Albert form or an 8-dimensional form with nontrivial Clifford invariant, isometric to \(q\otimes\theta\) with \(\theta\) a 1-fold Pfister form. Moreover, it is shown that non-excellent forms of height 2 and degree \(\leq 2\) can be parametrized by cohomological invariants. For forms of degree \(>2\) some partial results are obtained.
scientific article; zbMATH DE number 885621
scientific article