On perfect k-rational cuboids

Let K be an algebraic number field. A cuboid is said to be K-rational if its edges and face diagonals lie in K. A K-rational cuboid is said to be perfect if its body diagonal lies in K. The existence of perfect Q-rational cuboids is an unsolved problem. We prove here that there are infinitely many distinct cubic fields K such that a perfect K-rational cuboid exists; and that, for every integer n≥ 2, there is an algebraic number field K of degree n such that there exists a perfect K-rational cuboid.