Algebra & Number Theory

Integral trace forms associated to cubic extensions

Guillermo Mantilla-Soler

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Abstract

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

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

Dates
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
https://projecteuclid.org/euclid.ant/1513729571

Digital Object Identifier
doi:10.2140/ant.2010.4.681

Mathematical Reviews number (MathSciNet)
MR2728486

Zentralblatt MATH identifier
1201.11100

Subjects
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

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

Citation

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


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