Kvaternionové řády a kvadratické formy
| Thesis title in Czech: | Kvaternionové řády a kvadratické formy |
|---|---|
| Thesis title in English: | Quaternion orders and quadratic forms |
| Academic year of topic announcement: | 2023/2024 |
| Thesis type: | diploma thesis |
| Thesis language: | |
| 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: | 08.04.2024 |
| Date of assignment: | 08.04.2024 |
| Confirmed by Study dept. on: | 08.04.2024 |
| Guidelines |
| Examining the algebraic properties of quaternion algebras and quaternion orders provides a way to study the quadratic forms in four variables that correspond to the reduced norm, as seen in the classical quaternionic proof of Langrange's and Jacobi's four-square theorems due to Hurwitz. The student will use these tools to prove results about universality and about the number of representations by totally positive quaternary quadratic forms over totally real number fields, particularly focusing on providing a full quaternionic proof of Götzky's four-square theorem. He may also further consider a broader examination of the orders where such a method may succeed, building on the known enumeration of definite quaternion orders of class number 1 [KL]. |
| References |
| [De] J. I. Deutsch, A Non-classical Quadratic Form of Hessian Discriminant 4 is Universal Over Q(√5), Integers 16 (2016), A19.
[KL] M. Kirschmer, D. Lorch, Ternary quadratic forms over number fields with small class number, J. Number Theory 161 (2016), 343--361. [Vo] J. Voight, Quaternion Algebras, Springer Cham, 2021. |