MA40238 Number Theory

2014/15 Semester 1

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.

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.

Recommended Texts

• 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.

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.

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.

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.

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.