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# Elementary algebraic number theory, Kevät 2017

**Teacher:** Hui, Gao

**Scope:** 5 cr

**Type:** Advanced

**Teaching:**

**Topics: **In this course, we study some basic notions and classical theorems in algebraic number theory. As the title "algebraic" suggests, we will need to build some tools from abstract algebra to study numbers. In fact, to prove our main theorems, we will also use techniques from "geometry of numbers". In this course, we will try to explain some of the most fundamental ideas and techniques in number theory, yet in a basic and accessible way. Some of these techniques find applications in other branches of mathematics as well.

**Prerequisites: Algebra II.**

It is necessary to be fairly familiar with concepts like groups, rings, fields, modules.

We will also need some very elementary notions in topology (which can certainly be learnt during the course).

## News

- NOTE: This course will last just 1 period (7 weeks). There will not be any continuation in the 4th period.
- My personal homepage:
__https://sites.google.com/site/huigaomath/home__ - My email address: hui.gao@helsinki.fi My Office: Exactum A423

#### Teaching log:

week1: Noetherian rings and modules. (3.1 of Book). Chapter 1 of Book (modules over PID).

week2: Book, Section2.1 until almost end of 2.6. Exercise session on Friday.

week3: Section 2.7, 2.9. Then defined Dedekind domain. Had exercise session on Wednesday.

week4: finished Chapter 3. (Noetherian rings are already covered in week 1). Had exercise session on Monday.

week5: finishe proof of finiteness of class number; stated unit theorem. had exercise session on Mon and Fri.

week6, finished proof of unit theorem on Wed. Then start Zariski toplogy.

week7. finished Zariski topology, tensor product. Introduced roughly category, functor, sheaf, affine scheme. Talked about Fermat Last Theorem in the final lecture (almost 40 people attended the talk!)

## Teaching schedule

Monday 14:15-16:00, BK106 Wednesday 14:15-16:00, BK106

## Exams

There will not be exams.

There will be exercises. To pass the course, you need to score 50% on the total exercises.

There are no different grades, just passed or non-passed.

HOMEWORK POLICY:

You can discuss with people about the problems. But you

*must*write the answers on your own. (Do not copy from other people.)Late homework submissions will

**not**be accepted. (If it is something very serious, e.g., being sick for an entire week, then I will assign you some other work.)Even if you can not completely solve the problem, you can write down whatever that you get. Partial credits will be given.

You can submit your homework to me before deadline, either during class meetings, or emails (scans, pictures, or pdf files). Once submitted, no change is allowed.

## Course material

We will use __Pierre Samuel's book "Algebraic Theory of Numbers____"__ **(Chapter 1 to 4)** as a guiding book. (We will use materials from other sources as well)

We list some of the algebra tools which we develop in the course:

1. Modules over principle ideal rings

2. Finite extension of fields (and their structures)

3. Noetherian rings and Dedekind rings

The final goal in the course is to prove the "finiteness of class number" theorem (and the "unit theorem", if time permits).

A rough plan: 7 weeks

1Week: Modules over principle ideal rings

2Weeks: Finite extension of fields, norm, trace, discriminant

2Weeks: Noetherian rings and Dedekind rings

2Weeks: Dirichlet's "Finite ideal class group Theorem" and "Unit Theorem".

## Registration

Did you forget to register? What to do?

## Exercises

### Assignments

__Homework 1 PDF____file__Due time: Before 13:00, Jan 26 (Thursday)Due: Before 13:00, Jan 30 (Monday)__HW 2.pdf____HW 3 corrected.pdf__Due: Before 23:00, Feb 4 (Saturday)Due: Before 16:00, Feb 10(Friday)__HW4.pdf__Due: Before 16:00, Feb 15 (Wednesday)*HW 5.pdf**HW 6 corrected.pdf*Due: Before 16:00, Feb 22 (Wednesday)Due: Before 16:00, Feb 27(Monday)__HW7.pdf__*HW7 survey result.pdf*

### Exercise classes

Group | Day | Time | Room | Instructor |
---|---|---|---|---|

3rd period | Fri | 14:15-16:00 | BK106 | Hui Gao |

## Course feedback

Course feedback can be given at any point during the course. Click here.