Advanced Variable Topics in Mathematics - Math 191 Lec 1 - Fall 09

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Course: Math 191/Lecture 1, Fermat's Last Theorem: An Introduction to Algebraic Number Theory
Instructor: Jared Weinstein
My Office Hours: W 2:00 pm - 3:00 pm, Th 1:00 pm - 2:00 pm in MS 5230
Lectures: MWF 1:00 pm - 1:50 pm, MS 5233

This course is a friendly introduction to the theory of algebraic numbers, motivated by early attempts at solving Fermat's Last Theorem. This infamous theorem, posed by Pierre de Fermat in 1637, states that no perfect power (other than a square) can ever be the sum of two other (positive) perfect powers of the same exponent. We'll start by studying the basic number theory needed to solve Fermat's Last Theorem for certain exponents, as was done by Fermat himself, Euler, and Sophie Germain. Then we'll show how the study of the arithmetic of algebraic numbers -- numbers such as √ 2 and i which are the roots of polynomials with rational coefficients -- hold the key to a very powerful attack on FLT, as in the work of Kummer. Key concepts include: Gaussian integers, unique factorization property, Kummer's "ideal numbers", and regular primes.

There will be weekly assignments and a final project, which will be either a paper or an in-class presentation. Prerequisite: Linear Algebra. Some familiarity with proofs (as in Math 115 or 131) is strongly recommended.

This course is a little different from other upper-division mathematics courses. We will study the history of how mathematicians attacked one specific problem as much as we study the problem itself. By the end of this course, you will be familiar with the work of Fermat, Euler, Gauss, Germain, Legendre, Dirichlet, Kummer, Eisenstein, and others as it applies to Fermat's Last Theorem. You will learn a fair amount of elementary number theory. You will also be exposed to groups, rings, and fields, as you would in an algebra course, but from a "bottom-up" approach. Instead of defining these structures axiomatically, we will begin with the very simplest examples and then gradually expose you to more complex ones.

A Classical Introduction to Modern Number Theory, by Ireland and Rosen, Springer, 2nd. ed., 1998.

This book treats all the material we'll do in the lectures, and much more besides. Don't be intimidated that it is a graduate-level book! We're not going to cover every part of every chapter. I will try to be very specific about which pages you'll be responsible for.

During the first two weeks of the quarter, I would like to meet with each of you personally for about five minutes. We will discuss your background in mathematics and also your motivation for taking this course. This will help me tailor the lectures to your needs and abilities. These meetings will take place in MS 5230 and will be set up by e-mail.

There will be a problem set due in lecture every Friday, with the exception of the first day of instruction and Oct. 30. No late homework will be accepted, but I will be dropping the two lowest homework scores from your average. You are encouraged to collaborate with other students on the assignments but every solution you turn in must be in your own words.

The assignments:
HW1 HW1 Solutions
HW2 HW2 Solutions
HW3 HW3 Solutions
HW4 HW4 Solutions
HW5 HW5 Solutions
HW6 HW6 Solutions
HW7

As in other upper-division math courses, some of the exercises require justifying your answer. For these problems, a rigorous proof must be given for full credit. Solutions are graded for clarity as much as correctness. If you are unsure if your solution is sufficiently clear or correct, I am willing to check your work before the assignment is due. This may mean dropping by my office hours or sending me an e-mail. (Give me about a day to respond.)

There will be one take-home midterm exam, due Fri., Oct. 30. This is an open-book exam. You may use your notes, your textbook, or even outside sources, so long as you always cite what you use. Your solutions will be subject to the same standards of clarity as your homework assignments. You will not be allowed to work with other students on the midterm.

You will be required to write a final paper which gives a detailed account of some aspect of Fermat's Last Theorem beyond what was covered in the lectures. (I will provide a list of suggested topics and sources.) This paper will give you an opportunity to improve your technical communication skills as well as learn some new mathematics. It should be about 10 pages and will be due on Thurs., Dec. 10 (note new due date!). You may use LaTeX for the typesetting but it is not required; one alternative is to hand-write the equations and type everything else using an ordinary word processor. Let me know if you would like to learn how to use LaTeX -- if you continue in the sciences after this course, it will almost certainly come in handy.

I highly recommend completing a first draft of the final paper more than a week before the due date. If you do this, I can read through it and return it to you with comments for a revision.

I will compute an average as follows: 30% homework, 30% midterm exam, 40% final project. There won't be any curve, so your grade won't be adversely affected by the performance of your fellow students. To get a good grade this in this course, I recommend the following: Come to every lecture, work with other students on the problem sets, discuss the problem sets with me during office hours, and submit a first draft of your final paper to me for comments.

A short history of Fermat's Last Theorem.

The Wikipedia article on Fermat's Last Theorem is pretty decent.

The book Fundamentals of Number Theory by William LeVeque has considerable overlap with this course.

The book Modular Forms and Fermat's Last Theorem contains an actual proof of FLT within its nearly 600 pages. It's for serious students of number theory, no doubt, but leafing through it you can get an idea of the variety of methods involved. Algebra, group theory, geometry, and complex analysis all play a role.