Some Symmetric Pyramids with trivial Dehn invariant

TL;DR

This paper investigates symmetric pyramids in three-dimensional space, identifying Galois obstructions related to their Dehn invariants and establishing that only rational symmetric pyramids are scissor equivalent to a cube.

Contribution

It introduces conditions involving abelian Kummer extensions that determine when symmetric pyramids are scissor equivalent to a cube, extending previous classifications of rational tetrahedra.

Findings

Galois obstructions prevent certain symmetric pyramids from having zero Dehn invariant.

Only rational symmetric pyramids are scissor equivalent to a cube.

Conditions involve abelian properties of associated number field extensions.

Abstract

For some symmetric pyramids of $R^3$ , we find Galois obstruction for their Dehn invariant to be zero, i.e. for the pyramids to be scissor equivalent to a cube. These conditions are that some associated Kummer extensions of number fields must be abelian. This work can be viewed as a complement to [5]. Indeed, in [5], a classification of all rational tetrahedra is given, while we concentrate our attention on much more symmetric pyramids and prove that the only ones which are scissor equivalent to a cube are the rational ones.