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THE LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS OVER SIMPLEST CUBIC FIELDS

Part of: Algebraic number theory: global fields Forms and linear algebraic groups

Published online by Cambridge University Press:  06 October 2023

DANIEL GIL-MUÑOZ Open the ORCID record for DANIEL GIL-MUÑOZ [Opens in a new window]  and
MAGDALÉNA TINKOVÁ Open the ORCID record for MAGDALÉNA TINKOVÁ [Opens in a new window]
DANIEL GIL-MUÑOZ
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 00 Praha 8, Czech Republic e-mail: daniel.gil-munoz@mff.cuni.cz
MAGDALÉNA TINKOVÁ*
Affiliation:
Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, 160 00 Praha 6, Czech Republic and Institute of Analysis and Number Theory, TU Graz, Kopernikusgasse 24/II, 8010 Graz, Austria
*
e-mail: tinkova.magdalena@gmail.com
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Abstract

The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $\mathbb {Z}$-form) that is universal over K. We prove the nonexistence of universal $\mathbb {Z}$-forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.

Keywords

universal quadratic formtotally real number fieldadditively indecomposable integer

MSC classification

Primary: 11R16: Cubic and quartic extensions
Secondary: 11E12: Quadratic forms over global rings and fields 11R04: Algebraic numbers; rings of algebraic integers 11R80: Totally real fields

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 1 , August 2024 , pp. 77 - 89
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by Czech Science Foundation GAČR, grant 21-00420M, and by Charles University Research Centre program UNCE/SCI/022. The second author was supported by Czech Science Foundation GAČR, grant 22-11563O.

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