On the ring of \(13\times 13\) generic matrices. (Q1770524)

Assume \(F\) is a fixed field of characteristic 0. It is well known that the ring of \(r\) generic matrices of size \(n\) has no zero divisors. Let \(C_n\) be the centre of the generic division ring. A celebrated result of Procesi states that \(C_n\) is unirational over \(F\) of transcendence degree \((r-1)n^2+1\). Now fix \(r=2\) that is consider the ring of two generic matrices. If \(n\) is a prime power then \(C_n\) is known to be stably rational over \(F\) for \(n=2\), 3, 4, 5, and 7. When \(n=2\), 3, 4, it is rational. The paper under review considers the case \(n=13\). The main result of the paper is that there exists a field extension of degree 2 of \(C_{13}\) that is stably isomorphic to a certain field extension of degree 2 of a rational extension of the base field \(F\).