For a given field F of characteristic different from 2 and \(a,b,d\in F^*\) we construct an invariant \(\mathrm{inv}\) for an element \(D\in \,_2\mathrm{Br}(F(\sqrt{a},\sqrt{b},\sqrt{d})/F)\). This invariant takes value in the quotient group

Let k be a field, let \(k(\sqrt{a},\sqrt{b},\sqrt{d})/k\) be a triquadratic field extension. We apply the invariant \(\mathrm{inv}\) and a few deep results from algebraic geometry and K-theory to construct a field extension K/k with \(\mathrm{cd}_2 K=3\), and an indecomposable cross product algebra of exponent 2 with respect to the extension \(K(\sqrt{a},\sqrt{b},\sqrt{d})/K\). Using the invariant \(\mathrm{inv}\), we also prove the following odd degree descent statement: Assume \(D\in \,_2\mathrm{Br}(F)\), \(b,d\in F^*\), L/F is an odd degree extension. Assume also that \(D_{L(\sqrt{b},\sqrt{d})}=Q_{L(\sqrt{b},\sqrt{d})}\), where Q is a quaternion algebra defined over L. Then there exists a quaternion algebra \(\widetilde{Q}\) defined over F such that \(D_{F(\sqrt{b},\sqrt{d})}=\widetilde{Q}_{F(\sqrt{b},\sqrt{d})}\). As a consequence we get that if \(\phi \in I^2(F)\) is a form such that \({(\phi _{L(\sqrt{b},\sqrt{d})})}_{an}\) is defined over L, and \(\dim {(\phi _{L(\sqrt{b},\sqrt{d})})}_{an} =4\) , then \({(\phi _{F(\sqrt{b},\sqrt{d})})}_{an}\) is defined over F.

对于特征不同于 2 和\(a,b,d\in F^*\)的给定域F，我们为元素\(D\in \,_2\ )构造一个不变式\(\mathrm{inv}\) mathrm{Br}(F(\sqrt{a},\sqrt{b},\sqrt{d})/F)\)。这个不变量在商组中取值

设k是一个域，设\(k(\sqrt{a},\sqrt{b},\sqrt{d})/k\)是一个三二次域扩展。我们应用不变性\(\mathrm{inv}\)和代数几何和K理论的一些深层结果来构造场扩展K / k与\(\mathrm{cd}_2 K=3\)，以及指数 2 的不可分解叉积代数相对于扩展\(K(\sqrt{a},\sqrt{b},\sqrt{d})/K\)。使用不变式\(\mathrm{inv}\)，我们还证明了以下奇次下降语句：假设 \(D\in \,_2\mathrm{Br}(F)\) , \(b,d\in F^*\) , L/ F 是奇数度扩展。还假设 \(D_{L(\sqrt{b},\sqrt{d})}=Q_{L(\sqrt{b},\sqrt{d})}\)，其中 Q 是定义的四元数代数超过 L。那么存在 定义在F 上的四元数代数\(\widetilde{Q}\) 使得\(D_{F(\sqrt{b},\sqrt{d})}=\widetilde{Q}_{F(\ sqrt{b},\sqrt{d})}\)。因此我们得到如果\(\phi \in I^2(F)\)是这样的形式\({(\phi _{L(\sqrt{b},\sqrt{d})}) }_{an}\)定义在L 上，并且\(\dim {(\phi _{L(\sqrt{b},\sqrt{d})})}_{an} =4\)，然后 \({(\phi _{F(\sqrt{b},\sqrt{d})})}_{an}\)在F上定义。