Intermediate Jacobians and rationality over an arbitrary field

(Joint work with Olivier Benoist.) We develop a theory of intermediate Jacobians for geometrically rational threefolds over an arbitrary, not necessarily perfect, field. We deduce that a 3-dimensional smooth intersection of two quadrics is rational if and only if it contains a line. We thus obtain the first counterexamples to the Luroth problem that become rational after a purely inseparable extension of scalars.