Bilinear forms of dimension less than or equal to 5 and function fields of quadrics in characteristic 2 (Q2852452)

Let \(F\) be a field of characteristic \(2\). If \(B\) is a nondegenerate bilinear form over \(F\), one denotes by \(\widetilde{B}\) the associated diagonal (and thus totally singular) quadratic form given by \(\widetilde{B}(x)=B(x,x)\). The authors consider the question when such an anisotropic bilinear form \(B\) becomes isotropic over \(F(\psi)\), the function field of the projective quadric defined by the equation \(\psi=0\), where \(\psi\) is an anisotropic quadratic form over \(F\). This can only happen when \(\psi\) is totally singular, i.e., \(\psi\cong \widetilde{C}\) for another bilinear form \(C\). Continuing earlier work by the first author [J. Algebra 355, No. 1, 1--8 (2012; Zbl 1284.11062)] on the case where \(B\) is an Albert form, i.e., where \(B\) is \(6\)-dimensional with trivial discriminant, they now give a complete solution for \(3\leq \dim B\leq 5\). Crucial tools are the so-called norm field \(N_F(\varphi)\) of a diagonal quadratic form \(\varphi\) over \(F\), i.e., the field generated over \(F^2\) by products \(\alpha\beta\) of values \(\alpha\), \(\beta\) represented by \(\varphi\), and the norm degree \([N_F(\varphi):F^2]\), as introduced by the first author and the reviewer in [Trans. Am. Math. Soc. 356, No. 10, 4019--4053 (2004; Zbl 1116.11020)].