For smooth projective varieties over a field, the Brauer group is a birational invariant. It has been fruitful. For example, when "X" is also rationally connected over the complex numbers, the Brauer group of "X" is isomorphic to the torsion subgroup of the singular cohomology group "H"3("X", Z), which is therefore a birational invariant. Artin and Mumford used this description of the Brauer group to give the first example of a unirational variety "X" over C that is not stably rational (that is, no product of "X" with a projective space is rational).