Arithmetics of number fields and generalized continued fractions
| Thesis title in Czech: | Aritmetika číselných těles a zobecněné řetězové zlomky |
|---|---|
| Thesis title in English: | Arithmetics of number fields and generalized continued fractions |
| Key words: | číselná tělesa|nerozložitelné prvky|univerzální kvadratické formy|vícerozměrné řetězové zlomky|Pythagorovo číslo |
| English key words: | number fields|indecomposable integers|universal quadratic forms|mutidimensional continued fractions|Pythagoras number |
| Academic year of topic announcement: | 2016/2017 |
| 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: | 20.09.2017 |
| Date of assignment: | 20.09.2017 |
| Confirmed by Study dept. on: | 03.10.2017 |
| Date and time of defence: | 28.07.2021 16:00 |
| Date of electronic submission: | 18.05.2021 |
| Date of submission of printed version: | 18.05.2021 |
| Date of proceeded defence: | 28.07.2021 |
| Opponents: | Valentin Blomer |
| Andrew Earnest | |
| Guidelines |
| The aim of the thesis is to study the arithmetics of number fields, in particular the structure of totally positive additively indecomposable integers. In the case of real quadratic fields, these can be nicely characterized in terms of periodic continued fractions, which then has applications to the study of universal quadratic forms. However, the classical theory no longer works for higher degree number fields, so the student will investigate and develop other methods based on generalized continued fractions or numeration systems, e.g., the Jacobi-Perron algorithm or continued fractions with algebraic coefficients. |
| References |
| 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. Vítezslav Kala, Universal quadratic forms and elements of small norm in real quadratic fields, Bull. Aust. Math. Soc. 94 (2016), 7-14. Fritz Schweiger, Multidimensional Continued Fractions, Oxford University Press, 2000. Julien Bernat, Continued fractions and numeration in the Fibonacci base, Discrete Math. 306 (2006), no. 22, 2828–2850. Leon Bernstein, The Jacobi-Perron Algorithm Its Theory and Application, Lecture Notes in Mathematics, Volume 207 (1971). |