Three theorems on common splitting fields of central simple algebras
The following theorem generalizes results of \textit{A. A. Albert} [Bull. Am. Math. Soc. 37, 301-302 (1931; Zbl 0001.26701)], \textit{L. J. Risman} [Proc. Am. Math. Soc. 51, 35-36 (1975; Zbl 0309.16014)]: Theorem 1. Let \(A\) be a central simple algebra of prime degree \(p\) over a field \(F\). Given \(p-1\) central simple \(F\)-algebras \(\{B_i\}\) of degrees a power of \(p\), if each tensor product \(A\otimes B_i\) has zero-divisors, then there is a field extension \(E/F\) of degree \(\leq k\) splitting \(A\) and the \(B_i\), where \(k\) is the product of the degrees of the \(B_i\). -- For a second result define two central simple algebras \(A\), \(B\) of prime degree \(p\) over a field to be linked if the tensor product of any tensor powers of \(A\) and \(B\) has index \(\leq p\). If they have a common splitting field of finite degree not divisible by \(p^2\), they are called strongly linked. Now the author proves that for algebras \(A\), \(B\) of odd prime degree that are not strongly linked, the base field can be extended so that they become linked but not strongly linked. -- Given a prime \(p\) and \(x_1,\dots,x_n\in\text{Br}(F)\), the Brauer group of \(F\), let \(\alpha\colon\mathbb{Z}^n\to\mathbb{Z}\) be such that for any \(i_1,\dots,i_n\in\mathbb{Z}\) the Schur index \(\text{ind}(i_1x_1+\cdots+i_nx_n)\) is \(p^a\), where \(a=\alpha(i_1,\dots,i_n)\); then the elements \(x_1,\dots,x_n\) are said to have behaviour \(\alpha\) and \(\alpha\) is called a behaviour. For each behaviour \(\alpha\) an integer \(\beta\) is defined in terms of the Brauer-Severi varieties of the corresponding \(x_i\) and the author proves, for given \(p\), \(\alpha\), \(\beta\), (i) for any field \(F\), any \(x_1,\dots,x_n\in\text{Br}(F)\) with behaviour \(\alpha\) have a common splitting field \(E/F\) such that \(v_p([E:F])\leq\beta\) and (ii) there is a field \(F\) and \(x_1,\dots,x_n\in\text{Br}(F)\) with behaviour \(\alpha\) such that all their common splitting fields \(E/F\) satisfy \(v_p([E:F])\leq\beta\). The proof, using ideas from algebraic geometry, also leads to the construction of generic algebras with given behaviour.
scientific article; zbMATH DE number 1356473
scientific article