Elementary algebraic number theory, Kevät 2017


Teacher: Hui, Gao

Scope: 5 cr

Type: Advanced


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).


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


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.
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".


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Exercise classes

3rd periodFri14:15-16:00BK106Hui Gao

Course feedback

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