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Most totally real fields do not have universal forms or the Northcott property.

Nicolas Daans1,2, Vítězslav Kala1, Siu Hang Man1

  • 1Department of Algebra, Faculty of Mathematics and Physics, Charles University, Praha 8 186 75, Czech Republic.

Proceedings of the National Academy of Sciences of the United States of America
|May 16, 2025
PubMed
Summary

The study finds that most totally real fields do not have a universal quadratic form or the Northcott property. This is demonstrated using a theorem about square classes of totally positive units in quadratic lattices.

Keywords:
Northcott propertyinfinite extensionstotally positive unitstotally real fielduniversal quadratic form

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Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Quadratic Forms

Background:

  • Totally real fields are fundamental objects in algebraic number theory.
  • The constructible topology provides a framework for studying these fields.
  • Universal quadratic forms and the Northcott property are key concepts in classifying fields.

Purpose of the Study:

  • To investigate the prevalence of fields admitting a universal quadratic form or possessing the Northcott property within the space of totally real fields.
  • To determine if these properties are common or rare.

Main Methods:

  • Utilizing the constructible topology on the space of totally real fields.
  • Developing and applying a theorem concerning the number of square classes of totally positive units.
  • Analyzing quadratic lattices of varying ranks.

Main Results:

  • The set of totally real fields admitting a universal quadratic form is meager.
  • The set of totally real fields with the Northcott property is meager.
  • A key theorem relates these properties to the representation of square classes by quadratic lattices.

Conclusions:

  • Fields with universal quadratic forms or the Northcott property are rare in the space of totally real fields.
  • The findings provide insights into the structure and classification of totally real fields.