Decomposition of Mal'cev-Neumann division algebras with involution
We consider a crossed product \(A\) with center \(F\) and which is elementary abelian of exponent 2. Then \(A\) is endowed with an involution \(\sigma\) leaving the maximal subfield \(K\) elementwise invariant. On the other hand, we consider the Mal'cev-Neumann series division algebra associated to \(A\) and endow it with an involution \(\sigma^*\) associated to \(\sigma\). The crossed product generators of \(A\) and of the series division algebra can be chosen fixed by \(\sigma\) and \(\sigma^*\) respectively. This allows us to compute the discriminants of \(\sigma\) and \(\sigma^*\). Then we give necessary and sufficient conditions for the series fields to have a \(\sigma^*\)-invariant quaternion subalgebra or to be decomposable as a tensor product of \(\sigma^*\)-invariant quaternion algebras. Finally we show that if \(A = \text{End}_ FK\) then \(\sigma\) is the adjoint involution with respect to the quadratic form \(q(x) = \text{Tr}_{K/F}(\alpha x^ 2)\) for some \(\alpha\) in \(K\). In this case, if \(\alpha\) is such that \(N_{K/F}(\alpha)\) is in \(F^{*2}\) but \(N_{K/L}(\alpha)\) is not in \(L^{*2}\) for some quadratic subfield \(L\) of \(K\), then the series field is not a tensor product of \(\sigma^*\)-invariant quaternion algebras but the discriminant of \(\sigma^*\) is trivial. Moreover, if the Clifford algebra of \(q\) has index at least 4, the series field has no \(\sigma^*\)-invariant quaternion algebra. We give such an example where the discriminant of \(\sigma^*\) remains trivial.
scientific article; zbMATH DE number 703849
scientific article