Binomial squares in pure cubic number fields

Lemmermeyer, Franz

Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 691-704.

Soit $K=\mathbb{Q}\left(\omega \right)$, avec ${\omega }^3=m>1$ un nombre entier, un corps de nombres cubique. Nous montrons que les éléments $\alpha \in K^{\times }$ avec ${\alpha }^2=a-\omega$ (où $a$ est un nombre rationnel) forment un groupe qui est isomorphe au groupe des points rationnels de la courbe elliptique $E_m:y^2=x^3-m$. Nous démontrons aussi comment utiliser cette observation pour construire des extensions quadratiques non ramifiées de $K$.

Let $K=\mathbb{Q}\left(\omega \right)$, with ${\omega }^3=m$ a positive integer, be a pure cubic number field. We show that the elements $\alpha \in K^{\times }$ whose squares have the form $a-\omega$ for rational numbers $a$ form a group isomorphic to the group of rational points on the elliptic curve $E_m:y^2=x^3-m$. This result will allow us to construct unramified quadratic extensions of pure cubic number fields $K$.

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