Algebra & Number Theory
- Algebra Number Theory
- Volume 4, Number 6 (2010), 681-699.
Integral trace forms associated to cubic extensions
Given a nonzero integer , we know by Hermite’s Theorem that there exist only finitely many cubic number fields of discriminant . 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 as such a refinement. For a cubic field of fundamental discriminant we show the existence of an element in Bhargava’s class group such that is completely determined by . By using one of Bhargava’s composition laws, we show that is a complete invariant whenever is totally real and of fundamental discriminant.
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
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. https://projecteuclid.org/euclid.ant/1513729571