Bertrand's and Rodriguez Villegas' Conjecture for real quartic Galois extensions of the rationals

TL;DR

This paper proves an improved lower bound for the 1-norm of elements in the exterior square of units in Galois quartic extensions of the rationals, advancing the understanding of Bertrand's and Rodriguez Villegas' conjecture.

Contribution

It establishes a new lower bound of 1.134 for Galois quartic extensions, improving previous bounds for totally real cases.

Findings

Lower bound of 1.134 for Galois quartic extensions

Improved understanding of the conjecture in specific number fields

Extension of previous bounds for exterior square units

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. For the exterior square case of totally real quartic extensions of the rationals, Costa and Friedman gave a lower bound of 0.802. We prove that the bound can be improved to 1.134 when the extension is further assumed to be Galois.