Number Theory

Prerequisites:

Fundamental Algebra and Analysis I (or equivalent course covering abstract algebra, linear algebra, and basic topology)

Course description:

This course provides a comprehensive introduction to algebraic number theory following Neukirch's foundational approach. It is designed for graduate students and advanced undergraduates who intend to pursue research in number theory, arithmetic geometry, or related fields. The teaching objective is to enable students to master the fundamental concepts and techniques of algebraic number theory, including the arithmetic of number fields, Dedekind domains, ideal theory, and valuations. Topics focus on the structure of rings of integers, class groups, unit groups, ramification theory, and the theory of local fields leading to p-adic numbers. This course will employ a variety of teaching methods, including lectures, problem solving sessions, and independent reading.

Learning objectives:

By the end of this course, students will be able to:

• Analyze the structure of algebraic integers and rings of integers of number fields, computing norms, traces, discriminants, and integral bases.

• Work with Dedekind domains, proving unique factorization of ideals and computing class groups and class numbers.

• Apply Minkowski theory and the geometry of numbers to prove finiteness of the class number and Dirichlet's unit theorem.

• Determine the splitting behavior of primes in extensions of number fields, including ramification and inertia.

• Analyze the arithmetic of cyclotomic fields and understand classical applications such as quadratic reciprocity.

• Construct and work with completions, p-adic numbers, and local fields, applying Hensel's lemma and understanding the local-global perspective.

Detailed topics covered:

1. Algebraic integers: integral elements, integral closure, rings of integers, norms, traces, and discriminants.

2. Dedekind domains: unique factorization of ideals, fractional ideals, the ideal class group.

3. Geometry of numbers: lattices, Minkowski theory, finiteness of the class number.

4. Units: Dirichlet's unit theorem, regulators, S-units.

5. Ramification theory: splitting of primes in extensions, ramification and inertia, Hilbert theory.

6. Cyclotomic fields: arithmetic of cyclotomic extensions, quadratic reciprocity.

7. Valuations and absolute values: discrete valuations, archimedean and non-archimedean absolute values, Ostrowski's theorem.

8. Completions and local fields: p-adic numbers, Hensel's lemma, extensions of complete fields.

References:

[1] Neukirch J. - Algebraic Number Theory

[2] Fesenko I. - Core Topics in Number Theory I

[3] Dolce P. - Diophantine Approximation over the Real Line (Course notes)