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

Let $K=ℚ\left(\omega \right)$, with ${\omega }^{3}=m$ a positive integer, be a pure cubic number field. We show that the elements $\alpha \in {K}^{×}$ 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 $K$.

Soit $K=ℚ\left(\omega \right)$, avec ${\omega }^{3}=m>1$ un nombre entier, un corps de nombres cubique. Nous montrons que les éléments $\alpha \in {K}^{×}$ 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 $K$.

@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/

[1] P. Barrucand, H. Cohn, A rational genus, class number divisibility, and unit theory for pure cubic fields. J. Number Theory 2 (1970), 7–21. | MR 249398 | Zbl 0192.40001

[2] M. Bhargava, A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. ArXiv:1006.1002v2.

[3] G. Billing, Beiträge zur arithmetischen Theorie der ebenen kubischen Kurven vom Geschlecht Eins. Nova Acta Reg. Soc. Ups. (IV) 11 (1938). | Zbl 0018.05401

[4] H. Cohen, J. Martinet, Heuristics on class groups: some good primes are not too good. Math. Comp. 63 (1994), no. 207, 329–334. | MR 1226813 | Zbl 0827.11067

[5] H. Cohn, A classical invitation to algebraic numbers and class fields. Springer-Verlag, 1978. | MR 506156

[6] H. Eisenbeis, G. Frey, B. Ommerborn, Computation of the 2-rank of pure cubic fields. Math. Comp. 32 (1978), 559–569. | MR 480416 | Zbl 0385.12001

[7] L. Euler, Vollständige Anleitung zur Algebra (E387, E388). St. Petersburg, 1770.

[8] T. Honda, Pure cubic fields whose class numbers are multiples of three. J. Number Theory 3 (1971), 7–12. | MR 292795 | Zbl 0222.12004

[9] D. Husemöller, Elliptic Curves. 2nd ed., Springer-Verlag, 2004. | MR 2024529 | Zbl 0605.14032

[10] J.-L. Lagrange, Sur la solution des problèmes indéterminés du second degré. Mem. Acad. Sci. Berlin, 1769.

[11] J.-L. Lagrange, Additions à l’analyse indéterminée. Lyon, 1774.

[12] F. Lemmermeyer, A note on Pépin’s counter examples to the Hasse principle for curves of genus $1$. Abh. Math. Sem. Hamburg 69 (1999), 335–345. | MR 1722943 | Zbl 0949.11019

[13] F. Lemmermeyer, Why is the class number of $ℚ\left(\sqrt[3]{11}\phantom{\rule{0.166667em}{0ex}}\right)$ even? Math. Bohemica, to appear. | Zbl 1274.11162

[14] T. Nagell, Solution complète de quelques équations cubiques à deux indéterminées. J. Math. Pures Appl. 4 (1925), 209–270. | JFM 51.0135.02

[15] pari, available from http://pari.math.u-bordeaux.fr

[16] sage, available from http://sagemath.org

[17] P. Satgé, Un analogue du calcul de Heegner. Invent. Math. 87 (1987), 425–439. | MR 870738 | Zbl 0616.14023

[18] J. Silverman, J. Tate, Rational Points on Elliptic Curves. Springer-Verlag, 1992. | MR 1171452 | Zbl 0752.14034