Class numbers of orders in real quadratic fields

• Thesis title in Czech: Třídová čísla řádů v reálných kvadratických tělesech
• Thesis title in English: Class numbers of orders in real quadratic fields
• Key words: Kvadratické pole|Číselné pole|Třídová grupa|Řád|Relativní třídová grupa|Relativní třídové číslo|Pellová rovnice
• English key words: Quadratic field|Number field|Class group|Order|Relative class group|Relative class number|Pell's equation
• Academic year of topic announcement: 2024/2025
• Thesis type: Bachelor's thesis
• Thesis language: angličtina
• Department: Department of Algebra (32-KA)
• Supervisor: doc. Mgr. Vítězslav Kala, Ph.D.
• Author: Arina Beck - assigned and confirmed by the Study Dept.
• Date of registration: 10.03.2025
• Date of assignment: 10.03.2025
• Confirmed by Study dept. on: 10.03.2025
• Date and time of defence: 04.09.2025 09:00
• Date of electronic submission: 17.07.2025
• Date of submission of printed version: 17.07.2025
• Opponents: Subham Roy, Ph.D.

Guidelines

The student will prove Dirichlet's formula for the relative class number by carefully constructing a series of homomorphisms between factors of ideal class groups. Using this formula, she will consider the properties of orders in real quadratic fields of relative class number 1. Moreover, she will relate these to the Ankeny-Artin-Chowla and Mordell Conjectures.

References

Mak Trifcović (2013), Algebraic Theory of Quadratic Numbers, Springer Science+Business Media New York
Debopam Chakraborty and Anupam Saikia (2014), Another look at real quadratic fields of relative class number 1, Acta arithmetica 163.4
Andreas Reinhart (2024), A counterexample to the Pellian equation conjecture of Mordell, arxiv:2402.09827