# 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

Gaining mastery over an advanced area of algebraic number theory of interest in research.

### Prerequisites

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

### Minimal learning outcomes

These cannot be specified uniquely for a topics course. Past topics have included "Primes of the form *x ^{2}+Ny^{2}*," "The Proof of Fermat's Last Theorem," and "Computational Algebraic Number Theory." The following example gives a clear indication of the level of difficulty appropriate to the course. Students should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving results of suitable accessibility.

- Primes of the form
*x*^{2}+Ny^{2}- Classical Topics
- Equivalence of binary quadratic forms
- Composition of Binary quadratic forms
- Relation between binary quadratic forms and ideal class groups of quadratic fields
- Representations of integers by binary quadratic forms

- Class field theory
- Artin Reciprocity and the Artin Map
- The Existence and Conductor Theorems
- Cebotarev Density Theorem
- Identifying Ray Class Fields and Ring Class Fields

- Complex Analytic Techniques
- Elliptic Functions
- Properties of the Weierstrass p-function
- Invariants of Lattices
- Modular forms--definitions and properties
- Singular values of modular forms
- The modular equation
- Computing generators for ring class fields

- Classical Topics

### Textbooks

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

- D. Cox, Primes of the form x^2+Ny^2
- H. Cohen, A course in computational number theory

### Additional topics

None

### Courses for which this course is prerequisite

None