# Algebraic Number Theory

Math 523 - Fall 2022

## Announcements

This page contains some information that can be helpful before you register for the course. After the registration, we will be using odtuclass. You should follow odtuclass and check your emails regularly for important announcements during the semester.

A previous homepage: Math 523 - Spring 2018

## Course Objectives

Algebraic number theory originated in attempts for solving Diophantine equations, most notable of which is Fermat's last theorem, a conjecture that remained open for more than 300 years. The theory is now mature. Through the factorization of ideals into products of prime ideals in number fields, one obtains numerous applications to problems in number theory and other fields of mathematics and science. The aim of this course is to introduce the student to this theory and endow him/her with the techiques necessary to perform computations and carry out proofs in this area.

## Lectures and Office Hour

Lectures: Monday 13:40-14:30 and Wednesday 13:40-15:30 in M215.

Office hour: Thursday 9:40-10:30.

## Homeworks, Exams and Grading

Homeworks will be assigned on a regular basis and there will be 4-6 homework sets by the end of the semester. There will be one midterm and a final. **The time and the method of each exam will be announced later.**

Midterm, 30 points - around the 8th week.

Final, 30 points - during the final exam period.

Homework, 40 points.

**Homework Policy:** You should write your solutions on your own. You are allowed to consult other people's solutions for homework problems, but you must express everything in your own words. If you copy a solution, which is referred to as cheating, you will probably gain nothing and may encounter penalties.

## Textbooks and Tentative Course Outline

There are four textbooks. However we will mostly use the first two.

**1) Marcus, Number Fields,**

**2) Stewart & Tall, Algebraic Number Theory and F.L.T., 3rd edition**,

**3) Washington, Introduction to Cyclotomic Fields, 2nd edition** and

**4) Cox, Primes of the Form x ^{2} + ny^{2}.**

Find a tentative outline below for the whole semester. For each week, we will attempt to cover the indicated pages.

**(Oct 03 - Oct 07)** Introduction. Pythagorean triples and Gaussian
integers. [1, 1-11]

**(Oct 10 - Oct 14)** Algebraic background. [2, 9-34]

**(Oct 17 - Oct 21)** Algebraic numbers. Algebraic integers. Integral bases. [2, 35-48]

**(Oct 24 - Oct 28)** Traces. Norms. Rings of integers. [2, 49-59]

**(Oct 31 - Nov 04)** Cyclotomic fields. [1, 17-19, 27, 30-36]

**(Nov 07 - Nov 11)** Factorization into irreducibles. Euclidean domains. [2, 79-93]

**(Nov 14 - Nov 18)** Two Diophantine equations. Dedekind domains. [2, 94-98] [1, 55-62]

**(Nov 21 - Nov 25)** Prime factorization of ideals. [1, 62-82]

**Midterm** - TBA

**(Nov 28 - Dec 02)** Galois theory applied to prime decomposition. [1, 98-114]

**(Dec 05 - Dec 09)** Class group and class number. Minkowski's constant [1, 130-140].

**(Dec 12 - Dec 16) ** Class group examples.

**(Dec 19 - Dec 23)** Dirichlet's unit theorem. Pell's equation. [1, 141-146]

**(Dec 26 - Dec 30)** The first case of Fermat's Last Theorem. [3, 1-8]

**(Jan 02 - Jan 06)** Class number formula. Primes of the form x^{2}+ny^{2}. [3] and [4]

**Final** - TBA

## PARI / GP

The software PARI / GP is very simple to learn and

extremely strong to do computations with.

## A note for undergraduate students

If you are an undergraduate student, your CGPA must be higher than 3.00 in order to take this course. If you are willing to take this course, please send me an email before the registration.