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ON PERFECT
$K$-RATIONAL CUBOIDS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 04 October 2017, pp. 26-32
- Print publication:
- February 2018
-
- Article
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- You have access
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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.
