Composition of quadratic forms over number fields

• Thesis title in Czech: Skládání kvadratických forem nad číselnými tělesy
• Thesis title in English: Composition of quadratic forms over number fields
• Key words: binární kvadratické forma, třídová grupa, číselné těleso, Bhargavova krychle
• English key words: binary quadratic form, ideal class group, number field, Bhargava cube
• Academic year of topic announcement: 2017/2018
• Thesis type: diploma thesis
• Thesis language: angličtina
• Department: Department of Algebra (32-KA)
• Supervisor: doc. Mgr. Vítězslav Kala, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 04.10.2017
• Date of assignment: 04.10.2017
• Confirmed by Study dept. on: 27.11.2017
• Date and time of defence: 14.06.2018 09:00
• Date of electronic submission: 01.05.2018
• Date of submission of printed version: 11.05.2018
• Date of proceeded defence: 14.06.2018
• Opponents: Pavel Francírek

Guidelines

M. Bhargava recently found a new view on Gauss composition of binary quadratic forms through certain "cubes of integers". M. W. Mastropierto proved the correspondence between the ideal class group and binary quadratic forms (and hence the Gauss composition) over real quadratic fields of class number one. The goal of the thesis is to study and connect these two results: to generalize Gauss composition to more general number fields, and to possibly try to use it for a generalization of the composition of Bhargava cubes.

References

[1] Bhargava, M.: Higher composition laws I: A new view on Gauss composition, and quadratic generalizations, Ann. of Math., 159 (2004), pp. 217–250
[2] Mastropierto, M. W.: Quadratic Forms and Relative Quadratic Extensions, dissertation, 2000, https://pdfs.semanticscholar.org/e79e/def15345fdc48779f7187e2e6b3757d70081.pdf
[3] O'Dorney, E.: Rings of small rank over a Dedekind domain and their ideals, 2015, arXiv:1508.02777