Pete  L. Clark
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  Math 4400/6400 -- Number Theory (2009): MWF 11:15-12:05, 221 Boyd

Instructor: Assistant Professor Pete L. Clark, pete (at) math (at) uga (at) edu

Course webpage: (i.e., right here)

Office Hours: Boyd 502, X + Y (to be announced soon!), and by appointment.
I am quite amenable to booking ``extra'' office hours. The ground rules are: (i) please give me at least 24 hours notice. (ii) Please send me an email the night before a morning appointment or the morning of a later appointment to remind me that we are meeting. (iii) If I do not show for an appointment (empirically, the chance that I will fail to show seems to be about 5-10%), feel free to call me on my cell phone. Probably I'm not too far away. (iv) Please understand that I may be coming into campus specifically or primarily to meet you. I am more than happy to do this -- unless you don't show up. If you can't make it, please let me know ASAP.

Course text: You do not need to purchase any textbook for this class. In the course of teaching the class last time (in spring of 2007) I wrote up a rather extensive set of lecture notes (about 160 pages). Then in the intervening years I wrote up some further notes (about 75 more pages), on topics that I was sad not to be able to cover the first time around and also one that was inspired by Dr. Patrick Corn, who taught the course in the spring of 2008. Homework problems are listed separately. Thus there is, essentially, a textbook of moderate size freely available in pieces on this page and also on the old course page.

Discussion Section: I would like to have an extra discussion section, one hour a week, to discuss problems and also for students to present short presentations. I will send out email trying to find a minimally inconvenient time for this.

How Your Grade is Computed: Roughly (magic words: "This is a general plan. Deviations may be necessary.")
50% homework
20% final exam
20% final paper
20% in class (includes participation plus performance on a midterm exam).

More on Your Course Grade: Surely you noticed that above adds up to 110%. This gives an idea of the grading philosophy: any student who comes to almost all of the lectures, makes a point of turning in the homework with complete solutions to the easier problems and evidence of time spent thinking on the harder problems, whose performance on the exams shows mastery of comparatively elementary skills like solving linear Diophantine equations and computing Legendre symbols, and who turns in a coherent, original and somewhat thoughtful final paper will get a B- grade or higher. The expectation is not that you will understand every single thing that you are exposed to and solve all of the problems (there are some problems that I have only vague ideas about how to solve, some that no one yet knows how to solve, and one whose correct solution will earn you $1 million). Rather, the expectation is that you will be able to use the course to increase your understanding of number theory and related areas of mathematics, and also your interest and appreciation for the subject.

Prerequisites: This course offers an introduction to number theory, suitable for undergraduates majoring in mathematics. The main prerequisite is a course in modern algebra at the undergraduate level equivalent to (or greater than) our Math 4000, as well as what one would expect from any 4000-level class: a prior exposure to theoretical mathematics and mathematical proofs and some level of comfort and memory of previous courses like calculus and linear algebra.

In other words, Math 4400 is a course for math majors towards the end of their undergraduate career (or with equivalent knowledge and experience). It is possible -- and even somewhat traditional -- to teach a very elementary undergraduate number theory course, which would be equally accessible to bright high school students. This is not such a course. I assume for instance that you have had prior exposure not only to congruence modulo n, but to the notion of congruence modulo an ideal in a ring. Not that you have to have mastered such things seamlessly (whatever that might mean): rather there are handouts describing every algebraic concept and theorem that we will use, starting with the definition of a ring. In other words, I will be more than happy to remind and help you with abstract algebra, but it's much harder to remind someone of something that they never knew!

Graduate credit: Some of you are enrolled in Math 6400, a graduate course. There is only one set of lectures, so what's the difference? There are several things:
1) 6400 students are be expected to have a more solid and mature command of their background knowledge. This does not necessarily mean that they will be required to know more material, but they will have a graduate level thoroughness and flexibility with regards to the material that they do know.
2) 6400 students are expected to solve more of the difficult homework problems, including some which are more abstract and may involve some outside reading.
3) 6400 students will write longer and more substantial final papers. The papers will either be concerned with deeper mathematical topics or demonstrate a more sophisticated level of scholarship -- i.e., will synthesize material from multiple sources, including some primary sources.

Algebra Handouts:

  • Handout A1: Rings, Fields and (Mostly Commutative) Groups. (9 pages) (pdf)

  • Handout A2: Ideals and Quotients. (6 pages) (pdf)

  • Handout A2.5: More on Commutative Groups. (17 pages) (pdf)

  • Handout A3: Integral Elements and Extensions. (4 pages) (pdf)

    Handouts: Please note: as of the begining of January 2009, what appears below are the handouts from the 2007 version of this course. As our course progresses, this space will change. In particular, the ordering of the handouts may change, the individual handouts will undergo revisions and additions for 2009-style number theory, and we will certainly not be able to cover all of them. In case you are curious, here are my plans: we will certainly not cover any of the material in Handouts 17 or 18, except possibly Legendre's Theorem (Theorem 4 on page 5 of Handout 17). I definitely do wish to cover Bonus Handouts II and III. In order to make room, I am considering not covering Handouts 9.5 (which is an important result, but not one that we will need later on) and Handout 11 as well as removing some material from Handouts 9 and 10. In general, in 2007 I devoted a lot of time to elementary analytic number theory: that is, asymptotic estimates and inequalities. This time, for a switch, I will try to go a little deeper into algebraic considerations, especially more about the arithmetic of quadratic and cyclotomic fields.

    Part I: Introduction: Fundamental Theorem, Linear Diophantine Equations, Some Irrationality Proofs

  • Introduction; The Fundamental Theorem and Some Appplications. (17 pages) (pdf)

  • Some Irrational Numbers. (3 pages) (pdf)

    Part II: Rational and Integral Points on Curves

  • Pythagorean Triples. (10 pages) (pdf)

    Part III: Algebraic Number Theory

  • Quadratic Rings and Sums of Squares. (8 pages) (pdf)

  • Quadratic Reciprocity I. (12 pages) (pdf) NEW REVISION! PLEASE READ

  • Quadratic Reciprocity II: The Proof. (6 pages) (pdf)

  • The Pell Equation. (10 pages) (pdf)

    Part IV: Arithmetical Functions

  • Arithmetical Functions I: Multiplicative Functions. (6 pages) (pdf)

  • Arithmetical Functions II: Convolution and Inversion. (6 pages) (pdf)

  • Arithmetical Functions III: Orders of Magnitude. (8 pages) (pdf)

    Part V: The distribution of primes

  • The Primes: Infinitude, density and substance. (10 pages) (pdf)

  • The Prime Number Theorem and the Riemann Hypothesis. (8 pages) (pdf)

    Part VI: Geometry of Numbers

  • Gauss' Circle Problem. (8 pages) (pdf)

  • A Theorem of Minkowski; The Four Squares Theorem. (13 pages) (pdf)

    Part VII: The Chevalley-Warning Theorem

  • The Chevalley-Warning Theorem (Featuring...the Erdos-Ginzburg-Ziv Theorem). (14 pages) (pdf)

    Part VIII: Dirichlet series and Dirichlet's theorem It is rather likely that we will not get to this material.

  • Dirichlet Series. With the assistance of R. Francisco and D. May. (18 pages) (pdf)

  • Dirichlet's Theorem. (8 pages) (pdf)


    Homework Assignment 0: Due Wednesday, 1/21/09, further problems due Wednesday 1/28/09 (pdf)

    Homework Assignment 1: Due Monday, 2/9/09 (pdf)

    Homework Assignment 2: Due Wednesday, 2/18/09 (pdf)

    Homework Assignment 3: Due Friday, 2/27/09 (pdf)

    Homework Assignment 4: Due Monday, 3/23/09 (pdf)

    Homework Assignment 5: Due Monday, 3/30/09 (pdf)

    Homework Assignment 6: Due Wednesday, 4/8/09 (pdf)

    Homework Assignment 7: Due Friday, 4/24/09 (pdf)

    A list of 30 final project ideas, described in varying amounts of detail: (pdf)


    You (no matter who you are) are more than welcome to read, copy, print, or link to any part of these notes. I ask only the following:

    (i) If you wish to make use of any material from these notes, you do need to cite them explicitly. Failure to do so may constitute plagiarism.
    (ii) If you create any paper, note or webpage which uses and cites these notes, please let me know about it!

    So far, internet searches have turned up the following citations:

    Quadratic Reciprocity, Jordan Schettler, University of Arizona

    Quaternion algebras and quadratic forms, Master's thesis, Zi Yang Sham, University of Waterloo

    Ungelöste Rätsel, Ein kleiner mathematischer Rundgang, Jörg Resag

  • Davenport-Cassels Theorem, French Wikipedia

    Group Theory: an introduction and an application, Nathan Hatch

    VIGRE Seminar: Introducing the p-adic numbers, Jim Stankewicz, UGA (click on Writings/Talks and scroll to the bottom)

    Minkowski's Convex Body Theorem, Isabelle Bensimon, Boston University

    Project IV (MATH4072) 2013-14 The Polynomial Method, Dan Evans, Durham University