Cohomological invariants for central simple algebras of degree 8 and exponent 2

Abstract

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

$$\begin{aligned} H^3(F,\mu _2)/D\cup {\mathrm{N}_{\mathrm{F}\left( \sqrt{\mathrm{d}}, \sqrt{\mathrm{ab}}\right) /\mathrm{F}}}F\left( \sqrt{d},\sqrt{ab}\right) ^*. \end{aligned}$$

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.