Degree four cohomological invariants for certain central simple algebras (Q1732902)

The author introduces in this paper an invariant of bi-quaternion algebras \(A=(\alpha,\beta)_{2,F} \otimes (\gamma,\delta)_{2,F}\) over fields \(F\) of \(\operatorname{char}(F)\neq 2\) under the condition that \(-1\) is the sum of two squares in \(F\) and \((-1)\cup A\) is trivial as an element of \(H^3(F,\mu_2^{\otimes 3})\). This invariant is simply the symbol \((\alpha,\beta,\gamma,\delta)\) in \(H^4(F,\mu_2^{\otimes 4})\), and the proof relies on the chain lemma that the author provided in an earlier work [J. Algebra 350, No. 1, 170--173 (2012; Zbl 1253.16020)]. This invariant coincides with the familiar divided power operator when \(\sqrt{-1}\in F\). Another invariant the author introduces concerns sums \(C\otimes \omega\), where \(\omega \in Br_2(F)\) and \(C\) is a central simple algebra of degree dividing 4. This invariant is \((C^{\otimes 2})\cup \omega\) as an element of \(H^4(F,\mu_2^{\otimes 4})\).