Binomial Squares in Pure Cubic Number Fields

Abstract

Let K = Q(ω) with ω^3 = m be a pure cubic number field. We show that the elementsα\in K^\times whose squares have the form a - ωform 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.