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Academic Year 2023/2024 - Teacher: Marco D'ANNA

Expected Learning Outcomes

The aim of the course is to give to the student a first approach to algebraic number theory, using the notions learned in the course of Commutative Algebra and also other concepts in commutative ring theory and field theory. Moreover the aim of the course is to refine the capacity of abstraction of the student, showing how a theoretical approach allows to obtain interesting numerical results.

Course Structure

In the course the will be lectures and exercises, given at the blackboard by the lecturer, and class exercises. Ussually the lecturer alternates exercises and theoretical parts in the same day. As for class exercises, the lecturer gives some exercises to the students, that have to try to solve them working in small groups; the lecturer helps the students to find the proper way to appoach the exercises. 

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

Required Prerequisites

Knowledge of basic concepts in commutative algebra and field theory; in particular it is useful to know integral ring extensions and algebraic field extensions.

Detailed Course Content

Recap on integral ring extensions.

Norms, traces and discriminants.

Quadratic and cyclotomic extensions.

Dedekind domains.

Factorization of prime ideals in extensions.

The ideal class group.

The Dirichlet unit theorem.

If there will be time other extra contents (e.g. factorization of prime ideals in Galois extensions) will be added.

Textbook Information

1. Notes given by the lecturer.

2. Marcus D.A, Number Fields, Springer 1977

3. Stewart I and Tall D, Algebraic number theory, Chapman and Hall 1987 

Course Planning

 SubjectsText References
1Recap on integral extensions1
2Quadratic and cyclotomic extensions1
3Norms, traces and discriminants.1
4Dedekind domains1
5Factorization of prime ideals in extensions1
6The ideal class group1
7The Dirichlet unit theorem1