• rationality
• jacobian

This work is devoted to studying the k-rationality of threefolds over a perfect field k. In particular, we prove the following theorem: the three-dimensional smooth complete intersection X of two quadrics over a perfect base field k is k-rational if and only if it contains a line defined over k. This theorem was proved in the case of an arbitrary base field by O. Benoit and O. Wittenberg in their article "Intermediate Jacobians and Rationality over arbitrary Fields". The assumption that the base field is perfect allows us to use the theory of intermediate jacobians (defined by Murre over an algebraically closed field). We explore and expand this theory, showing that Murre's intermediate jacobian defined over an algebraic closure of a perfect field k descends (by Galois descent) to an abelian variety over k and to torsors under the action of such abelian variety. The study of these objects yields some obstructions to the k-rationality of smooth projective threefolds that are crucial in the proof of the main theorem.