Counting extensions of imaginary quadratic fields
| Thesis title in Czech: | Počítání rozšíření imaginárních kvadratických těles |
|---|---|
| Thesis title in English: | Counting extensions of imaginary quadratic fields |
| Key words: | algebraická teorie čísel, teorie třídových těles, Cohen-Lenstrova heuristika |
| English key words: | algebraic number theory, class field theory, Cohen-Lenstra heuristics |
| Academic year of topic announcement: | 2019/2020 |
| Thesis type: | Bachelor's 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: | 20.12.2019 |
| Date of assignment: | 20.12.2019 |
| Confirmed by Study dept. on: | 11.02.2020 |
| Date and time of defence: | 22.09.2020 09:00 |
| Date of electronic submission: | 04.06.2020 |
| Date of submission of printed version: | 04.06.2020 |
| Date of proceeded defence: | 22.09.2020 |
| Opponents: | Pavlo Yatsyna, Ph.D. |
| Guidelines |
| The far-reaching Cohen-Lenstra heuristics concern asymptotics of number fields with prescribed properties (about their Galois and class groups). The goal of the thesis is to work out some special cases, namely to determine the asymptotics for the number of quadratic extensions of a given imaginary quadratic field K (with odd class number).
This requires counting characters of the absolute Galois group of K which, by class field theory, correspond to characters of the idele group of K. Their number is then determined from the corresponding L-functions using a Tauberian theorem. The student will formulate the necessary preliminaries and carry out these calculations partly following [1] which dealt with a similar problem for extensions of rational numbers. |
| References |
| [1] Melanie Matchett Wood: Asymptotics for number fields and class groups, http://swc.math.arizona.edu/aws/2014/2014WoodNotes.pdf
[2] J. S. Milne: Algebraic Number Theory, http://www.jmilne.org/math/CourseNotes/ant.html [3] J. S. Milne: Class Field Theory, http://www.jmilne.org/math/CourseNotes/cft.html [4] S. Lang: Algebraic Number Theory, GTM 110 |