Algebra & Number Theory

Integral trace forms associated to cubic extensions

Guillermo Mantilla-Soler

Full-text: Open access


Given a nonzero integer d, we know by Hermite’s Theorem that there exist only finitely many cubic number fields of discriminant d. However, it can happen that two nonisomorphic cubic fields have the same discriminant. It is thus natural to ask whether there are natural refinements of the discriminant which completely determine the isomorphism class of the cubic field. Here we consider the trace form qK: trK(x2)|OK0 as such a refinement. For a cubic field of fundamental discriminant d we show the existence of an element TK in Bhargava’s class group  Cl(222;3d) such that qK is completely determined by TK. By using one of Bhargava’s composition laws, we show that qK is a complete invariant whenever K is totally real and of fundamental discriminant.

Article information

Algebra Number Theory, Volume 4, Number 6 (2010), 681-699.

Received: 18 June 2009
Revised: 5 December 2009
Accepted: 15 May 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E12: Quadratic forms over global rings and fields
Secondary: 11R29: Class numbers, class groups, discriminants 11R16: Cubic and quartic extensions 11E76: Forms of degree higher than two

integral trace forms cubic fields Bhargava's class group discriminants of number fields


Mantilla-Soler, Guillermo. Integral trace forms associated to cubic extensions. Algebra Number Theory 4 (2010), no. 6, 681--699. doi:10.2140/ant.2010.4.681.

Export citation


  • K. Belabas and H. Cohen, “Binary cubic forms and cubic number fields”, pp. 191–219 in Computational perspectives on number theory (Chicago, 1995), edited by D. A. Buell and J. T. Teitelbaum, AMS/IP Stud. Adv. Math. 7, Amer. Math. Soc., Providence, 1998.
  • M. Bhargava, “Higher composition laws, I: A new view on Gauss composition, and quadratic generalizations”, Ann. of Math. $(2)$ 159:1 (2004), 217–250.
  • D. A. Buell, Binary quadratic forms, Springer, New York, 1989.
  • P. E. Conner and R. Perlis, A survey of trace forms of algebraic number fields, Series in Pure Mathematics 2, World Scientific, Singapore, 1984. 0551.10017
  • B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs 10, Amer. Math. Soc., Providence, 1964.
  • G. Eisenstein, “Théorèmes sur les formes cubiques et solution d'une équation du quatrième degré à quatre indéterminées”, J. Reine Angew. Math. 27 (1844), 75–79.
  • V. Ennola and R. Turunen, “On totally real cubic fields”, Math. Comp. 44:170 (1985), 495–518.
  • W. T. Gan, B. Gross, and G. Savin, “Fourier coefficients of modular forms on $G\sb 2$”, Duke Math. J. 115:1 (2002), 105–169.
  • H. Hasse, “Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage”, Math. Z. 31:1 (1930), 565–582.
  • D. Hilbert, “Theorie der algebraischen Zahlkörper”, Encykl. d. math. Wiss. 1 (1900), 675–714.
  • J. W. Hoffman and J. Morales, “Arithmetic of binary cubic forms”, Enseign. Math. $(2)$ 46:1-2 (2000), 61–94.
  • D. A. Marcus, Number fields, Springer, New York, 1977. 0383.12001 \xoxZBL
  • A. Scholz, “Über die Beziehung der Klassenzahlen quadratischer Körper zueinander”, J. Reine Angew. Math. 166 (1932), 201–203.