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.