Number theory I.a - Algebraic number theory
Summary
Algebraic number theory is the study of the properties of solutions of polynomial equations with integral coefficients; Starting with concrete problems, we then introduce more general notions like algebraic number fields, algebraic integers, units, ideal class groups...
Content
- Basics on rings and modules especially over a PID, lattices in R^n
- Dedekind rings
- The ring of integers of a number field is a Dedekind ring.
- Finiteness of the ideal class group
- the action of the Galois group.
- Dirichlet's units theorem
- Applications
The course will make use of concepts and techniques borrowed from other courses (some taking place during the same semester) such as Ring and Modules, Analytic NT and Galois theory.
As such it can be considered representative of the Diversity and Inclusivity inherent to mathematics.
Keywords
Rings, Fields, integers, ideals, lattices
Learning Prerequisites
Required courses
MATH-215
Recommended courses
MATH-311
MATH-313
MATH-317
Learning Outcomes
By the end of the course, the student must be able to:
- Quote the main results of the course
- Use the main results of the course
- Prove the main results of the course
Teaching methods
ex-cathedra
Expected student activities
attendance to the course and active participation to the exercises sessions
Assessment methods
written exam
Supervision
| Assistants | Yes |
| Others | moodle page |
Resources
Virtual desktop infrastructure (VDI)
No
Notes/Handbook
a pdf of the course will be provided
Websites
Moodle Link
Prerequisite for
MATH-417
MATH-489
MATH-494
Fields medal
Dans les plans d'études
- Semestre: Automne
- Forme de l'examen: Ecrit (session d'hiver)
- Matière examinée: Number theory I.a - Algebraic number theory
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel