# Math 686R: Topics in Algebraic Number Theory.

## Contents

## Catalog Information

### Title

Topics in Algebraic Number Theory.

### Credit Hours

3

### Prerequisite

### Description

Current topics of research interest.

## Desired Learning Outcomes

To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.

### Prerequisites

A sound understanding of basic concepts of complex analysis, abstract algebra and number theory, at the level of Math 352, 371, 372 and 487 may suffice.

### Minimal learning outcomes

1. Generalization of the Unique Factorization Theorem from rationals to number fields: Basic Definitions - number fields, algebraic integers in a number field, integral bases, discriminant, norms of ideals, finiteness of ideals of bounded norm, class number, finiteness of class number, Dedekind's Unique Factorization Theorem for ideals of a number field. 2. Geometry of numbers: Minkowski's lemma on lattice points, Logarithmic spaces, Dirichlet's Unit Theorem for the units of the ring of integers of a number field, theorems of Minkowski and of Hermite on discriminants of number fields. 3. Ramification Theory: Relative extensions, relative discriminant and Dedekind's criterion for ramification in terms of discriminant, higher ramification groups and Hilbert theory of ramification. 4 Splitting of Primes: Frobenius map, Artin symbol, Artin map and splitting of primes in Abelian extensions in terms of Artin map, rudimentary class field theory, Examples - quadratic and cyclotomic extensions. 5. Arithmetic of cyclotomic fields, the Kronecker-Weber Theorem. 6. Dedekind zeta function, the class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant.

- Generalization of the Unique Factorization Theorem from rationals to number fields: Basic Definitions - number fields, algebraic integers in a number field, integral bases, discriminant, norms of ideals, finiteness of ideals of bounded norm, class number, finiteness of class number, Dedekind's Unique Factorization Theorem for ideals of a number field.
- Geometry of numbers: Minkowski's lemma on lattice points, Logarithmic spaces, Dirichlet's Unit Theorem for the units of the ring of integers of a number field, theorems of Minkowski and of Hermite on discriminants of number fields.
- Ramification Theory: Relative extensions, relative discriminant and Dedekind's criterion for ramification in terms of discriminant, higher ramification groups and Hilbert theory of ramification.
- Splitting of Primes: Frobenius map, Artin symbol, Artin map and splitting of primes in Abelian extensions in terms of Artin map, rudimentary class field theory, Examples - quadratic and cyclotomic extensions.
- Arithmetic of cyclotomic fields, the Kronecker-Weber Theorem.
- Dedekind zeta function, the class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant.

In addition, time permitting, the instructor may want to add to the list topics of special to him/her.

### Textbooks

Possible textbooks for this course include (but are not limited to):