Let , with a positive integer, be a pure cubic number field. We show that the elements whose squares have the form for rational numbers form a group isomorphic to the group of rational points on the elliptic curve . This result will allow us to construct unramified quadratic extensions of pure cubic number fields .
Soit , avec un nombre entier, un corps de nombres cubique. Nous montrons que les éléments avec (où est un nombre rationnel) forment un groupe qui est isomorphe au groupe des points rationnels de la courbe elliptique . Nous démontrons aussi comment utiliser cette observation pour construire des extensions quadratiques non ramifiées de .
@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}, 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 = {http://archive.numdam.org/articles/10.5802/jtnb.817/} }
TY - JOUR AU - Lemmermeyer, Franz TI - Binomial squares in pure cubic number fields JO - Journal de théorie des nombres de Bordeaux PY - 2012 SP - 691 EP - 704 VL - 24 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.817/ DO - 10.5802/jtnb.817 LA - en ID - JTNB_2012__24_3_691_0 ER -
%0 Journal Article %A Lemmermeyer, Franz %T Binomial squares in pure cubic number fields %J Journal de théorie des nombres de Bordeaux %D 2012 %P 691-704 %V 24 %N 3 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.817/ %R 10.5802/jtnb.817 %G en %F 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://archive.numdam.org/articles/10.5802/jtnb.817/
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