# Algebraic Number Theory

Math 523 - Fall 2021

## 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.

The previous homepage: Math 523 - Spring 2018

## Lectures and Office Hours

Lectures will be face to face in M203 (Tosun Terzioglu Seminar Room) during the following times:

Tuesday 9:40-11:20, M203.

Thursday 11:40-13:20, M203.

The last hour (Thursday 12:40-13:20) is the problem session and it is optional. It will be held if you are willing to work on problems related with algebraic number theory.

## Homeworks, Exams and Grading

Homeworks will be assigned on a regular basis and there will be 5-7 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 with 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 18 - Oct 22)** Introduction. Pythagorean triples and Gaussian
integers. [1, 1-11]

**(Oct 25 - Oct 29)** Algebraic background. [2, 9-34]

**(Nov 01 - Nov 05)** Algebraic numbers. Algebraic integers. Integral bases. [2, 35-48]

**(Nov 08 - Nov 12)** Traces. Norms. Rings of integers. [2, 49-59]

**(Nov 15 - Nov 19)** Cyclotomic fields. [1, 17-19, 27, 30-36]

**(Nov 22 - Nov 26)** Factorization into irreducibles. Euclidean domains. [2, 79-93]

**(Nov 29 - Dec 03)** Two Diophantine equations. Dedekind domains. [2, 94-98] [1, 55-62]

**(Dec 06 - Dec 10)** Prime factorization of ideals. [1, 62-82]

**Midterm** - TBA

**(Dec 13 - Dec 17)** Galois theory applied to prime decomposition. [1, 98-114]

**(Dec 20 - Dec 24)** Class group and class number. Minkowski's constant [1, 130-140].

**(Dec 27 - Dec 31) ** Class group examples.

**(Jan 03 - Jan 07)** Dirichlet's unit theorem. Pell's equation. [1, 141-146]

**(Jan 10 - Jan 14)** The first case of Fermat's Last Theorem. [3, 1-8]

**(Jan 17 - Jan 21)** 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.