Generators of central simple algebras
Let \(A\) be a central simple algebra with centre \(K\). Given a subfield \(k\) of \(K\) and \(z_1,\dots,z_m\in A\), the subalgebra \(k(z_1,\dots,z_m)\) of \(A\) is defined by taking the \(k\)-algebra \(B\) generated by \(z_1,\dots,z_m\) and localizing at all elements of \(B\cap K^\times\). The author asks which algebras can be generated in this way, for a given \(m\), and in answer proves the following theorem. Let \(A\) be a central simple algebra of degree \(n\) over its centre \(K\). Given an integer \(m\geq 2\), suppose that \(K\) is a finitely generated extension field of \(k\) such that \(\text{tr.deg}_k(K)\leq(m-1)n^2\). Then \(A=k(z_1,\dots,z_m)\) for some \(z_1,\dots,z_m\in A\). He notes the special cases \(n=1\), giving a form of the primitive element theorem and the case \(\text{tr.deg}_k(K)\leq 4\), showing that a noncommutative algebra can be generated by two elements over \(k\). He gives two proofs, a fairly direct one for characteristic zero and one that is valid generally. He also notes that the result is a converse of a theorem of \textit{C. Procesi} [Rings with polynomial identities, Dekker, New York (1973; Zbl 0262.16018), VIII. 3].
scientific article; zbMATH DE number 1213804