Binomial squares in pure cubic number fields

Lemmermeyer, Franz

doi:10.5802/jtnb.817

Lemmermeyer, Franz ¹

¹ Mörikeweg 1 73489 Jagstzell Germany

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

Résumé
Abstract

Soit $K = ℚ (ω)$ , avec $ω^{3} = m > 1$ un nombre entier, un corps de nombres cubique. Nous montrons que les éléments $α \in K^{\times}$ avec $α^{2} = a - ω$ (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 $K = ℚ (ω)$ , with $ω^{3} = m$ a positive integer, be a pure cubic number field. We show that the elements $α \in K^{\times}$ whose squares have the form $a - ω$ 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 $K$ .

MR Zbl

DOI : 10.5802/jtnb.817

Affiliations des auteurs :

Lemmermeyer, Franz ¹

¹ Mörikeweg 1 73489 Jagstzell Germany

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     title = {Binomial squares in pure cubic number fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {691--704},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {3},
     year = {2012},
     doi = {10.5802/jtnb.817},
     zbl = {1269.11108},
     mrnumber = {3010635},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jtnb.817/}
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Lemmermeyer, Franz. Binomial squares in pure cubic number fields. Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 691-704. doi : 10.5802/jtnb.817. https://www.numdam.org/articles/10.5802/jtnb.817/

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