Binomial squares in pure cubic number fields
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, p. 691-704

Let K=(ω), with ω 3 =m a positive integer, be a pure cubic number field. We show that the elements αK × 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.

Soit K=(ω), avec ω 3 =m>1 un nombre entier, un corps de nombres cubique. Nous montrons que les éléments αK × 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.

@article{JTNB_2012__24_3_691_0,
     author = {Lemmermeyer, Franz},
     title = {Binomial squares in pure cubic number fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {3},
     year = {2012},
     pages = {691-704},
     doi = {10.5802/jtnb.817},
     mrnumber = {3010635},
     zbl = {1269.11108},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2012__24_3_691_0}
}
Lemmermeyer, Franz. Binomial squares in pure cubic number fields. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 691-704. doi : 10.5802/jtnb.817. http://www.numdam.org/item/JTNB_2012__24_3_691_0/

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