Published online by Cambridge University Press: 04 October 2017
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
$\mathbb{Q}$ -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
$n\geq 2$ , there is an algebraic number field
$K$ of degree
$n$ such that there exists a perfect
$K$ -rational cuboid.