- To learn about the arithmetic of algebraic number fields
- To prove theorems about integral bases, and about unique factorisation into ideals
- To prove the finiteness of the class number and calculate some examples
- To use the theory to solve simple Diophantine equations
The course gives an introduction to algebraic number theory, with a focus on examples such as quadratic number fields and also cyclotomic number fields. After a review of some key algebraic concepts, we introduce rings of integers of number fields, and norms and traces of algebraic numbers. We study prime factorisation in rings of integers, which unlike for the usual integers, are not necessarily unique. We define Dedekind domains and study unique factorisation of ideals. We introduce the ideal class group and the class number, and prove the finiteness of the class number.|
One of the guiding motivations for the theory was to solve Diophantine equations (the most famous example being Fermat's Last Theorem) and we will also see some applications in this direction.
|Ringen en Lichamen, Groepentheorie|
|Schriftelijk tentamen + opgaven|