# Math 686R: Topics in Algebraic Number Theory.

## Contents

## Catalog Information

### Title

Topics in Algebraic Number Theory.

### Credit Hours

3

### Prerequisite

Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.

### Description

Current topics of research interest.

## Desired Learning Outcomes

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

### Prerequisites

The pre-requisites in the Catalog are adequate. Depending on the instructor and with her/his permission, a sound understanding of basic concepts of complex analysis, abstract algebra and number theory might be adequate, perhaps at the level of Math 352, 371, 372 and 487.

### Minimal learning outcomes

- 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
- Hilbert theory of ramification

- Splitting of Primes:
- Frobenius map
- Artin symbol
- Artin map
- Splitting of primes in Abelian extensions in terms of Artin map
- Rudimentary class field theory
- Examples - quadratic and cyclotomic extensions

- Arithmetic of cyclotomic fields, and 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 interest to him/her.

### Textbooks

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

- D. Marcus, Number Fields (Universitext)