Unit Reducible Fields and Perfect Unary Forms

TL;DR

This paper introduces the concept of unit reducibility in number fields, explores its relationship with perfect unary forms, and proves a conjecture regarding the classification of these forms in real quadratic fields.

Contribution

It defines unit reducibility for number fields, links it to perfect unary forms, and proves an open conjecture about their classification in real quadratic fields.

Findings

Defined unit reducibility for number fields.

Established a connection between unit reducibility and perfect unary forms.

Proved an open conjecture on the number of classes of perfect unary forms in real quadratic fields.

Abstract

In this paper, we introduce the notion of unit reducibility for number fields, that is, number fields in which all positive unary forms attain their nonzero minimum at a unit. Furthermore, we investigate the link between unit reducibility and the number of homothety classes of perfect unary forms for a given number field, and prove an open conjecture about the number of classes of perfect unary forms in real quadratic fields, stated by D. Yasaki.