On indecomposable algebras of exponent \(2\).
For every integer \(n\geq 3\), the author constructs a central division algebra of degree \(2^n\) (i.e., dimension~\(4^n\)) and exponent~\(2\) (and characteristic different from~\(2\)) that does not decompose into a tensor product of proper subalgebras, even after an odd-degree extension of the base field. The algebra is Brauer-equivalent to a tensor product of \(n+1\) quaternion algebras \(D\otimes(a_1,t_1)\otimes\cdots\otimes(a_n,t_n)\) over a field of iterated Laurent series \(F(\!(t_1)\!)\cdots(\!(t_n)\!)\), where \(D\) is a suitably constructed quaternion \(F\)-algebra split by the multiquadratic extension \(L=F(\sqrt{a_1},\dots,\sqrt{a_n})\) of degree~\(2^n\). This quaternion algebra \(D\) was first considered by the author [in J. Algebra 275, No. 2, 859-866 (2004; Zbl 1133.11027)]. Its Brauer class is not the sum of \(2\)-torsion elements split by proper subextensions, so \(_2\text{Br}(L/F)\neq{_2\text{Br}}(L_1/F)+{_2\text{Br}}(L_2/F)\) for any \(L_1,L_2\subsetneq L\). A similar property holds for relative Witt groups, using the norm form of \(D\). The first examples of indecomposable division algebras of degree~\(8\) and exponent~\(2\) were constructed by \textit{S. A. Amitsur, L. H. Rowen} and \textit{J.-P. Tignol}, [Isr. J. Math. 33, 133-148 (1979; Zbl 0422.16010)]. They are Brauer-equivalent to tensor products of four quaternion algebras by \textit{J.-P. Tignol}, [C. R. Acad. Sci., Paris, Sér. A 286, 875-876 (1978; Zbl 0374.16006)].
scientific article; zbMATH DE number 5320447