Nonexcellence of multiquadratic field extensions
A field extension \(L/F\) is said to be excellent if for any quadratic form \(f\) over \(F\) the anisotropic part of \(f\) over \(L\) is defined over \(F\). Quadratic extensions of \(F\) are known to be excellent and if all multiquadratic extensions of \(F\) are excellent, then Merkurjev's theorem on the injectivity of Clifford invariant follows in an elementary way (that is, with no use of algebraic geometry or \(K\)-theory). For this see \textit{R. Elman} et al. [Proc. Conf. Quadratic Forms, Kingston 1976, Queen's Pap. Pure Appl. Math. 46, 445--492 (1977; Zbl 0385.12010)]. However, it was soon discovered that biquadratic extensions can be nonexcellent, the first counterexamples constructed by \textit{R. Elman} et al. [Am. J. Math. 105, 1119--1170 (1983; Zbl 0492.10014)]. The present author proves that, for any field \(k_0\) of characteristic not \(2\), if \(k_0\) has \(n+1\) multiplicatively independent square classes \(a,b_1,\ldots,b_n\), then there exists a field extension \(F/k_0\) with the property that the multiquadratic extension \(K=F(\sqrt{b_1}, \ldots, \sqrt{b_n})\) of \(F\) is not excellent. A 4-dimensional form \(f\) over \(F\) of determinant \(a\) is constructed, which is isotropic over \(K\) and anisotropic over any subfield of \(K\), and the anisotropic part of \(f\) over \(K\) is not defined over \(F\).
scientific article; zbMATH DE number 2105401
scientific article