On the tensor product of division algebras
Let \(F_ 0\) be a field containing \(\xi\) a primitive n-th root of unity and \(F_ 1:=F_ 0(X,Y,Z)\) for X,Y,Z variables over \(F_ 0\). Let \(F:=F_ 1((t_ 1))((t_ r))\) be the Laurent series field in two variables. Consider the following two division algebras \(D_ 1\) and \(D_ 2:\) \(D_ 1:=(X(1-Y),t_ 1)_{n,F}\otimes (X(1-Z),t_ 1)_{n,F}\text{ and } D_ 2:=(X,Y,Z)_{n,F}\)where, as usual, for \(a,b\in F^*\) the symbol \((a,b)_{n,F}\) denotes the cyclic algebra generated by two elements i,j such that \(i^ n=a\); \(j^ n=b\) and \(ij=\xi ji.\) In this note it is proved that \(index(D_ 1\otimes D_ 2^{\otimes n})<index(D_ 1)\cdot index(D_ 2)\) for all \(n<\exp onent\)(D\({}_ 2)\) but nevertheless \(D_ 1\) and \(D_ 2\) have no common subfield properly containing F. This result gives a negative answer to a question of Rowen, see \textit{L. H. Rowen}'s book: Ring theory, vol. II (Academic Press, 1988; Zbl 0651.16002), p. 217. The above construction also gives a negative answer to the following question of Max Knus: Let \(A_ 1,A_ 2,A_ 3\) be three quaternion division algebras over a field f. If \(A_ 1\otimes_ FA_ 2\otimes_ FL\) is a division algebra for any quadratic extension L of F contained in \(A_ 3\), is \(A_ 1\otimes A_ 2\otimes A_ 3\) a division algebra?
scientific article; zbMATH DE number 4171152
scientific article