Universal quadratic forms over orders in number fields
| Thesis title in Czech: | Univerzální kvadratické formy nad řády v číselných tělesech |
|---|---|
| Thesis title in English: | Universal quadratic forms over orders in number fields |
| Key words: | kvadratická forma|číselné těleso|součet čtverců|Pythagorovo číslo|kvadratický Waringův problém |
| English key words: | quadratic form|number field|sum of squares|Pythagoras number|quadratic Waring's problem |
| Academic year of topic announcement: | 2017/2018 |
| Thesis type: | dissertation |
| 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: | 27.09.2018 |
| Date of assignment: | 27.09.2018 |
| Confirmed by Study dept. on: | 29.10.2018 |
| Date and time of defence: | 13.09.2023 10:00 |
| Date of electronic submission: | 02.05.2023 |
| Date of submission of printed version: | 04.07.2023 |
| Date of proceeded defence: | 13.09.2023 |
| Opponents: | prof. Gabriele Nebe |
| Karim Johannes Becher | |
| Guidelines |
| The goal of the thesis is to study universal and regular quadratic forms over orders in totally real number fields and other closely related topics. Most of the recent research concerning indecomposable integers and the ranks of universal forms has been done only over the maximal order in a real quadratic field. One of the aims of the thesis is to extend these results to general orders. This is closely related to the study of quadratic forms that are assumed to represent only some arithmetic progression of algebraic integers, and to regular quadratic forms. |
| References |
| Juergen Neukirch, Algebraic Number Theory, Springer 1999.
Oskar Perron, Die Lehre von den Kettenbruechen, B.G. Teubner, 1913. David A. Cox, Primes of the Form x^2+ny^2: Fermat, Class Field Theory, and Complex Multiplication, Wiley, 1989. Wai Kiu Chan and Maria Ines Icaza, Positive definite almost regular ternary quadratic forms over totally real number fields, Bull. London Math. Soc., 40 (2008), 1025-1037. Valentin Blomer and Vitezslav Kala, Number fields without n-ary universal quadratic forms, Math. Proc. Cambridge Philos. Soc. 159 (2015), no. 2, 239–252. Vítezslav Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207. Valentin Blomer and Vitezslav Kala, On the rank of universal quadratic forms over real quadratic fields, Doc. Math. 2018. |