A fundamental question in birational geometry is determining if a given variety is (stably) rational or not. One of the useful invariants to distinguish (stably) irrational varieties from (stably) rational ones is the existence of a decompositon of the diagonal. Especially after the degeneration methods introduced by Voisin in 2015, there has been much progress using this invariant. However, one can often only prove that the very general member of a family is not stably rational. In this talk we explain how unramified cohomology and decomposition of the diagonal can be used to prove the stable irrationality of a specific intersection of a quadric and a cubic fivefold given by explicit equations. The main step in the argument is a degeneration to positive characteristic.