**Graduate Number Theory**

Mathematics 676: spring 2007

** Lecture:** MWF 1:10-2:00, Engineering E206. code 354250

**Prerequisite:**
Math 566 or permission of professor.

**Course description:**
The study of number theory originated in ancient civilizations
such as those of China and India and was developed in great depth in Europe
in the 17th and 18th centuries.
Number theory is known for having problems that are easy to state yet which
can only be solved using complicated structures.
For example, it took 300 years to find a complete proof of Fermat's Last Theorem.
Number theory is a subject that's intertwined with group theory, algebraic
geometry, combinatorics, and complex analysis. It's become popular recently
because of its applications to coding theory and cryptography.

Number theory is a vast subject. In this course, we will emphasize its algebraic and geometric aspects.
Here are some of the possible themes of the course.
In the first week of class, we will have an introduction to these topics and
choose which subset of them to cover.

1) Reciprocity Laws: the quadratic reciprocity law (which has over 100 proofs) tells you
whether or not a number is a square modulo a prime.

2) Ideal Factorization: this topic helps you measure the failure of unique
factorization in quadratic integer rings. The proof relies on
Minkowski's theorem on the geometry of lattices in the plane. These lattices
are used in recent cryptosystems.

3) Riemann zeta function: this function measures how primes are distributed
among the integers. The Clay Mathematics Institute is offering $1,000,000
for the the solution to the Riemann hypothesis.

4) Diophantine equations: this topic is about solutions to equations with
coordinates in fields like Q or F_p. It includes the topic of elliptic
curves (again with applications to cryptography) and Fermat's Last Theorem.
Some of the best data-transfer codes rely on this topic.

**Homework:**
**
Goals and syllabus **

We will cover about 2/3 of the following syllabus.
The course grade will be based 50% on homework and 50% on the projects.

** A. Algebraic Number Fields:**

1. Ideal factorization, especially in quadratic and cyclotomic integer rings;

2. Ramification theory;

3. Minkowski theory of lattices (plus applications to cryptosystems);

4. Finiteness of the ideal class group;

5. Dirichlet's Unit Theorem.

** B. Reciprocity Laws: **

1. Quadratic Reciprocity;

2. Kronecker-Weber Theorem.

** C. Diophantine Equations:**

1. Pell's equation;

2. Introduction to Fermat's last theorem;

3. Elliptic curves;

4. Applications to error-correcting codes;

5. Introduction to zeta functions.

** D. Distribution of Prime Numbers:**

1. Density and distribution of primes.

2. Introduction to the Riemann-zeta function.

**Texts:**
Fortunately, many excellent textbooks contain this material.
It is hard to recommend one in particular, but I picked Janusz as a main text.
Some books are expensive but all of them should be in the library.
We will talk about textbook choices in the first week of class.

Algebraic Number Fields 2nd edition, G. Janusz, Graduate Studies in Mathematics, Volume 7, AMS $45

clearly written, Chapter 1, pg 1-82 for topic A and B1.

A Classical Approach to Modern Number Theory, 2nd edition, Ireland-Rosen, Springer-Verlag $70

clearly written, best text for several topics but does not contain others

contains more background material, good problems but not many examples

Chapters 12-13 pg 172-202 for topic A. Chapters 5,7 (9?) for topic B1. Chapters 17 and 11 for topic C. Chapters 2 and 16 for topic D.

Classical Theory of Algebraic Numbers, Ribenboim $80.

long-winded approach, many examples, uninspiring

Chapters 4-14 (15.1? 16?) pg 61-333 for topic A. Part 4 for topic D.

Problems in Algebraic Number Theory, Esmonde and Murty, Springer $60.

contains many problems and their solutions.

Chapters 4-6, 8 for topic A. Chapter 7 (9?) for topic B. Chapter 10 for topic D.

Algebraic Number Theory, Neukirch, Springer $120

beautifully written, more sophisticated approach, not many examples

Chapter 1 pg 1-65 for topic A. Chapter 7 pg 419-443 for topic D.

Number Fields, D. Marcus, Springer

harder to read (typeset), harder approach

Chapters 2-5 for topic A. Chapter 1 for topic C2. Chapters 7-8 for topic D.

**Grading: ** This is an advanced graduate course and course grades will be computed accordingly.

60% Homework; 40% Project.

**Homework:**
**
Detailed information on homework**

** Project:**

The project is an opportunity to learn more about a topic in number theory that interests you or will be
relevant for your future graduate work.
It gives us a chance to hear about cool ideas which we will not have time to cover in class.
It is also a good opportunity to develop more skill at writing and speaking on mathematics.

**Help:** Help is always available if you have trouble with homework
or lecture material. If your classmates can't answer your question,
come ask me! Office hours will be (TBA) or are available by appointment.