The smallest regulator for number fields of degree 7 with five real places

TL;DR

This paper determines the minimal regulator for degree 7 number fields with five real places, extending geometric methods to include fields with one complex place, filling a gap in previous bounds.

Contribution

It extends Pohst's geometric method to handle degree 7 fields with five real places and identifies the field with the minimal regulator for this signature.

Findings

Identified the field with the first discriminant as having minimal regulator.

Extended Pohst's geometric method to fields with one complex place.

Provided sharp lower bounds for regulators of degree 7 fields with five real places.

Abstract

In 2016 Astudillo, Diaz y Diaz and Friedman published sharp lower bounds for regulators of number fields of all signatures up to degree seven, except for fields of degree seven having five real places. We deal with this signature, proving that the field with the first discriminant has minimal regulator. The new element in the proof is an extension of Pohst's geometric method from the totally real case to fields having one complex place.