Binomial Squares in Pure Cubic Number Fields

TL;DR

This paper explores the structure of elements in pure cubic number fields whose squares have a specific form, linking them to rational points on elliptic curves, and applies these findings to construct unramified quadratic extensions.

Contribution

It establishes an isomorphism between a subgroup of elements in pure cubic fields and rational points on elliptic curves, and demonstrates their application in constructing unramified quadratic extensions.

Findings

Elements with squares of form a - ω form a group isomorphic to elliptic curve points

Method to construct unramified quadratic extensions of pure cubic fields

Connection between algebraic elements and elliptic curve rational points

Abstract

Let K = Q(\omega) with \omega^3 = m be a pure cubic number field. We show that the elements\alpha \in K^\times whose squares have the form a - \omega form a group isomorphic to the group of rational points on the elliptic curve E_m: y^2= x^3 - m. We also show how to apply these results to the construction of unramified quadratic extensions of pure cubic number fields.