Bertrand's and Rodriguez Villegas' conjecture for real multi-quadratic Galois extensions of the rationals

TL;DR

This paper proves Bertrand's and Rodriguez Villegas' conjecture for the exterior square case in real multi-quadratic Galois extensions of the rationals, establishing a lower bound on the 1-norm of units.

Contribution

It provides the first proof of the conjecture for the exterior square case in a broad class of number fields, expanding understanding of units in Galois extensions.

Findings

Confirmed the conjecture for real multi-quadratic Galois extensions

Established a lower bound on the 1-norm of units in these fields

Extended the conjecture's validity to a new class of number fields

Abstract

The conjecture due to Bertrand and Rodriguez Villegas asserts that the 1-norm of the nonzero element in an exterior power of the units of a number field has a certain lower bound. We prove this conjecture for the exterior square case when the number field is a real multi-quadratic Galois extension of any degree of the rationals.