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Updated: May 7, 2026
07:46
Setting Limits on Supersymmetry Using Simplified Models
Published on: November 16, 2013
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.
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.
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.
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