MA40238 Number Theory

2014/15 Semester 1

Unit Description
Lectures and Office Hours
Recommended Texts
Calendar
Exercise Sheets Exam Information
Lecture Notes
Resources

Unit Description

This unit provides an introduction to number theory for students who have some background of abstract algebra. You should have taken Algebra 2B (MA20217 or equivalent) before taking this unit. Some familiarity of groups, rings and fields will be assumed. See resources for lecture notes of Algebra 2B in the last two years. You should also be comfortable with reading and writing rigorous mathematical proofs.

In the first half of the semester we will study elementary number theory. Topics to be covered in this part include: unique factorisation, arithmetic functions, Möbius inversion, congruences, Chinese remainder theorem, primitive roots, quadratic residues and quadratic reciprocity. You might have seen many of them in other units. However we will study these topics in a more comprehensive way.

In the second half of the semester we will introduce number fields. Topics to be covered in this part include: number fields, the ring of integers, traces and norms, unique factorisation of ideas, class numbers, ideal class groups and Minkowski theorem.

Here is the syllabus on the Unit Catalogue. A handout on course information is given out in the first lecture.

Lectures and Office Hours

The class meets three times a week. In general, Monday and Tuesday lectures are devoted to introducing new material, while Friday lectures will mainly focus on problem solving (except the first week). See course calendar for more details on the contents of each lecture.

Your lecturer for this unit is Dr. Ziyu Zhang. You can contact me by email at zz505 at Bath for any questions or concerns. I hold two office hour sessions each week. You are welcome to come and ask questions at any time during the sessions.

There will be extra office hours after the Christmas vacation. See exam information for more details on the time and location.


Monday
Tuesday
Wednesday Thursday Friday
09:15-10:05
Lecture
4W 1.7




11:15-12:05




Lecture
3E 3.5
15:15-16:05

Lecture
3E 3.5



16:30-17:30
Office Hour
4W 4.8


Office Hour
4W 4.8


Recommended Texts

The lectures do not follow any particular textbook. However the following two books cover most of the topics that we are discussing.
  • Ireland, Kenneth; Rosen, Michael. A classical introduction to modern number theory. Graduate Texts in Mathematics, 84. Springer-Verlag, New York-Berlin, 1982.
  • Marcus, Daniel. Number fields. Universitext. Springer-Verlag, New York-Heidelberg, 1977.
There are also many other good books in the library which are relevant to this unit. See the library catalogue.

Calendar

More than any other discipline, mathematics requires that a student understands one concept before moving on to the next. Therefore it is critical that you attend this class on a regular basis in order to stay on track with the material we cover. Brief lecture notes will be available here usually after each lecture for your convenience. Although these notes roughly reflect what is being discussed in lectures, reading these notes cannot replace attending lectures yourself.

You can find a complete set of lecture notes including all exercise sheets and solutions under the section of Lecture Notes.

Week 1
29/09
Welcome. Unique factorisation. Handout on course information.
Notes
30/09
Some arithmetic functions.
Notes
03/10
Möbius inversion theorem.
Notes
Week 2
06/10
Solving linear congruences.
Notes
07/10
Chinese remainder theorem.
Notes
10/10
Discussion of Exercise Sheet 1.

Week 3
13/10
Primitive roots for primes.
Notes
14/10
Primitive roots for powers of odd primes.
Notes
17/10
Discussion of Exercise Sheet 2. Handout on primitive roots.

Week 4
20/10
Quadratic residues. Legendre symbols.
Notes
21/10
Jacobi symbols.
Notes
24/10
Discussion of Exercise Sheet 3. Informal questionnaire.

Week 5
27/10
Gauss' lemma. Overview of the week.
Notes
28/10
A proof of quadratic reciprocity. Extra hints for Sheet 5.
Notes
31/10
Discussion of Exercise Sheet 4. Feedback for questionnaire.

Week 6
03/11
Algebraic numbers and algebraic integers. Overview of the week.
Notes
04/11
Number fields. Extra hints for Sheet 6.
Notes
07/11
Discussion of Exercise Sheet 5.
Week 7
10/11
The ring of integers. Overview of the week.
Notes
11/11
Integral bases of ideals. Extra hints for Sheet 7.
Notes
14/11
Discussion of Exercise Sheet 6.
Week 8
17/11
Finiteness of quotient rings. Overview of the week.
Notes
18/11
Unique factorisation of ideals. Extra hints for Sheet 8.
Notes
21/11
Discussion of Exercise Sheet 7.
Week 9
24/11
The ideal class group. Overview of the week.
Notes
25/11
Minkowski's theorem. Extra hints for Sheet 9.
Notes
28/11
Discussion of Exercise Sheet 8.
Week 10
01/12
Minkowski bound. Overview of the week.
Notes
02/12
Computing class numbers. Extra hints for Sheet 10.
Notes
05/12
Discussion of Exercise Sheet 9.
Week 11
08/12
Review. Handout on exam information.

09/12
Discussion of Exercise Sheet 10.

12/12
Non-examinable lecture: some aspects of Fermat's Last Theorem.


Exercise Sheets

There will be 10 exercise sheets in total. Every exercise sheet contains 4 problems, most of which have multiple questions. These problems are usually given in increasing difficulty, with the first one involving routine computations and the last one comprehensive and sometimes slightly challenging. Exercise sheets are designed to reinforce concepts covered in lecture as well as to encourage students to explore implications of the results discussed in class. Very few students will be able to go through the entire course without struggling on some problems, so do not be discouraged if you do not immediately know how to solve a problem. In confronting difficult questions you should consider how the problem at hand connects to topics, definitions and/or theorems discussed in class.

Although it is not mandatory to submit your work on a regular basis, it is strongly recommended that your try to work as much as possible on the exercises. If you have worked on a problem for a while and remain stuck, you are encouraged to discuss it with one another, or ask me for hints.  However if you have taken notes while discussing exercises with classmates, it will be a good idea to put these notes away when writing your own solution. Be warned: watching someone else solve a problem will not make exercises a good preparation for your exam. Writing down your own solution will help you think through the material and get a better understanding.

Exercise sheets are usually handed out on Tuesday and due in lecture on the following Tuesday. Please write your solutions neatly, with enough justification in each step. Don't forget to write your full name on your work. If it contains multiple pages, it will be helpful if you staple them before submission. At least two problems will be marked every week, but you should treat all four problems equally important. Submitted work will be handed back to you and discussed in the exercise class on Friday. Brief solutions will be posted here shortly after that.

Exercises Due Date
Solutions
Sheet 1
07/10
Solution 1
Sheet 2
14/10
Solution 2
Sheet 3 21/10
Solution 3
Sheet 4 28/10
Solution 4
Sheet 5 04/11
Solution 5
Sheet 6 11/11
Solution 6
Sheet 7
18/11
Solution 7
Sheet 8 25/11
Solution 8
Sheet 9
02/12
Solution 9
Sheet 10 N/A
Solution 10


Exam Information

  • Here is a handout on exam information.
  • Here is a mock exam with complete solutions.
  • Here are library copies of past exam papers. Notice that the syllabus has changed since last time this unit was given. So Problem 1 (a) and Problem 4 in the 2012/13 exam are no longer valid for this year. Here are brief solutions to the 2012/13 exam.
  • We will have one review session and a few office hour sessions before the exam. See the following chart for details. Please also feel free to contact me if you have questions at any time.


Monday
Tuesday
Wednesday
Thursday
Friday
Week of
5th-9th January




Review Session
3E 3.5
11:15-12:05
Week of
12th-16th January
Office Hour
4W 4.8
16:30-17:30


Office Hour
4W 4.8
16:30-17:30

Week of
19th-23rd January
Office Hour
4W 4.8
16:30-17:30
Office Hour
4W 4.8
16:30-17:30
Office Hour
4W 4.8
16:30-17:30
Office Hour
4W 4.8
16:30-17:30
Good Luck
with
Your Exam!


Lecture Notes

  • Here is a complete version of the lecture notes, exercise sheets and solutions for the whole course, with clickable hyperlinks in color and a table of contents. 107 pages.
  • Here is a print-friendly version, which contains the same material as the complete version, but with smaller fonts, page margins and line spacings. The table of contents and big chunks of empty spaces are also removed. 70 pages. More economic for printing purpose.

Resources

  • Here are the lecture notes of Algebra 2B (MA20217) in Semester 2 of 2012/13.
  • Here are the lecture notes of Algebra 2B (MA20217) in Semester 2 of 2013/14 on its webpage.
  • Here is the exam feedback which contains some general comments and common mistakes.

Department of Mathematical Sciences. University of Bath.
Maintained by Ziyu Zhang. Last update on Sunday, 1 March 2015.