pic from Scholze-Weinstein notes

In this topic course, we will learn some elementary algebraic number theory.

A diamond (recently invented by Peter Scholze) is a certain geometric object in current day algebraic number theory.

Enrol
8.10.2018 at 09:00 - 11.12.2018 at 23:59

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Hui Gao's picture

Hui Gao

Published, 10.12.2018 at 18:55

Survey result of "What other (algebra related) things are you interested to learn?"
Galois thy, 3 counts (also relation with topo space)
algebraic geometry, 2 counts
Representation thy
Class field theory
logic
algebraic study of symmetries
homological algebra

Timetable

Here is the course’s teaching schedule. Check the description for possible other schedules.

DateTimeLocation
Mon 29.10.2018
16:15 - 18:00
Tue 30.10.2018
16:15 - 18:00
Fri 2.11.2018
16:15 - 18:00
Mon 5.11.2018
16:15 - 18:00
Tue 6.11.2018
16:15 - 18:00
Fri 9.11.2018
16:15 - 18:00
Mon 12.11.2018
16:15 - 18:00
Tue 13.11.2018
16:15 - 18:00
Fri 16.11.2018
16:15 - 18:00
Mon 19.11.2018
16:15 - 18:00
Tue 20.11.2018
16:15 - 18:00
Fri 23.11.2018
16:15 - 18:00
Mon 26.11.2018
16:15 - 18:00
Tue 27.11.2018
16:15 - 18:00
Fri 30.11.2018
16:15 - 18:00
Mon 3.12.2018
16:15 - 18:00
Tue 4.12.2018
16:15 - 18:00
Fri 7.12.2018
16:15 - 18:00
Mon 10.12.2018
16:15 - 18:00
Tue 11.12.2018
16:15 - 18:00
Fri 14.12.2018
16:15 - 18:00

Material

Tasks

HW 1, due 6pm, Nov. 5

HW 2, due 6pm, Nov 12

HW 3, due 6pm, Nov. 19

HW 4, due 6pm, Nov. 26

HW 5, due 6pm, Dec. 3

HW 6, due 6pm, Dec. 10

Description

This course is suitable for advanced undergraduate students and master students.

The only Prerequisite: Algebra II.

It is necessary to be fairly familiar with concepts like groups, rings, fields, modules.
It is desirable if you know (or just heard of) some very basic Galois theory, but that is not required.
We will also need some very elementary notions in topology (which can certainly be learnt during the course).
The course would be rather similar to a course that I taught in 2017 (clickable link), but not entirely the same.

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.

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).
NOTE: This course will last just 1 period (7 weeks). There will not be any continuation of it.

Course material

We will use Pierre Samuel's book (the old version in 1970) "Algebraic Theory of Numbers" (Chapters 1 to 4 only) as a guiding book. (We will use materials from other sources as well)

The course book that we use is not available in university library. It is a rather old book (published in 1970), and is difficult to find. Here I put a link to find a pdf scan (I found it a year ago from a professor's course webpage, but the link disappeared). Please use it for this course only, and do not distribute the link.

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.
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 (a single pdf file). Once submitted, no change is allowed.

My email address: hui.gao@helsinki.fi
My Office: Exactum A423
My personal homepage: https://sites.google.com/site/huigaomath/home