Introduction to Algebraic Number Theory

2024/2025

Academic Year

ENG

Instruction in English

6

ECTS credits

Category 'Best Course for Career Development'

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Delivered at:

Faculty of Mathematics

Course type:

Optional course (faculty)

When:

1, 2 module

Instructor

Zhgoon, Vladimir

Course Syllabus

Abstract

Many classical and modern problems in number theory can be interpreted in terms of properties of algebraic objects such as algebraic number fields, their rings of integers and orders in these rings, ideal class groups and unit groups. In this course, we will explore the main notions of algebraic number theory and connect them to some of the most classical problems and theorems, for example, the Dirichlet theorem on primes in arithmetic progressions, Gauss class number problem and Fermat’s last theorem. We will also learn about the properties of analytic and topological objects corresponding to number fields, such as the Dedekind zeta-function and the ring of adeles.

Learning Objectives

• The course is intended to present the basics of modern algebraic number theory: properties of number fields and their rings of integers, structure of ideals, local fields, classical and adelic appoach to the Dedekind zeta-function.

Expected Learning Outcomes

• At the end of the course, students will be able to apply techniques of algebraic number theory to solve certain types of Diophantine equations and obtain qualitative and quantitative information about number fields

Course Contents

• Galois theory, finite fields, quadratic reciprocity law. Local fields, Ostrowski's theorem
• Theory of quadratic forms, Dirichlet characters. Dirichlet theorem on primes in arithmetic progressions. ∗Riemann zeta function and Prime Number Theorem
• Algebraic number fields. Trace and norm, different and discriminant. Dedekind domains. ∗Galois group of a typical polynomial
• Cyclotomic fields, Fermat’s last theorem, ∗regular and irregular primes
• Units and ideal classes. Dirichlet’s unit theorem. Ideal class group, Minkowski’s bound. ∗Class number formula
• Adeles and ideles, strong approximation. Dedekind zeta-function and its functional equation. ∗Extended Riemann hypothesis and Chebotarev density theorem.

Assessment Elements

• Sheets
• Exam

Interim Assessment

• 2024/2025 2nd module
  1/10(2/3(final exam%)+1/2(problem sheets%))

Bibliography

Recommended Core Bibliography

• Borevich, Z. I., & Shafarevich, I. R. (1966). Number Theory. Academic Press.
• Serge Lang. (2013). Algebraic Number Theory (Vol. 2nd ed. 1994). Springer.

Recommended Additional Bibliography

• Andre Weil. (2013). Basic Number Theory (Vol. 2nd ed. 1973). Springer.
• J-P. Serre. (2012). A Course in Arithmetic (Vol. 1973). Springer.

Authors

• ZHGUN VLADIMIR SERGEEVICH
• Иконописцева Юлия Вахтаногвна
• KALMYNIN ALEKSANDR Borisovich