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2010 Integral trace forms associated to cubic extensions
Guillermo Mantilla-Soler
Algebra Number Theory 4(6): 681-699 (2010). DOI: 10.2140/ant.2010.4.681


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.


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Guillermo Mantilla-Soler. "Integral trace forms associated to cubic extensions." Algebra Number Theory 4 (6) 681 - 699, 2010.


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

zbMATH: 1201.11100
MathSciNet: MR2728486
Digital Object Identifier: 10.2140/ant.2010.4.681

Primary: 11E12
Secondary: 11E76, 11R16, 11R29

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.4 • No. 6 • 2010
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