  We prove that a three-dimensional smooth complete intersection of two
quadrics over a field k is k-rational if and only if it contains a line defined
over k. To do so, we develop a theory of intermediate Jacobians for
geometrically rational threefolds over arbitrary, not necessarily perfect,
fields. As a consequence, we obtain the first examples of smooth projective
varieties over a field k which have a k-point, and are rational over a purely
inseparable field extension of k, but not over k.
