Instructor: Allen Hatcher

• Office: 553 Malott Hall
• Phone: 255-4091
• Email:
• Office Hours: Tues 4-5 pm, Wed 2-4 pm, and by appointment -- send me an email.
• TA: Ho Hon Leung. Office Hours Tues 3-4 pm, Thurs 12-1 pm.

Textbook: We will be using a book I'm in the process of writing, available as a free download here.

Exams: There will be two exams:

• Prelim: Thursday October 29, 7:30-9:00 pm. Covering the material up to page 25 in Chapter 2 of the book. No external assistance will be allowed during the exam: No notes or books, no electronic devices. Here is the exam with solutions.
• Final Exam: Friday December 18, 2:00-4:30 pm. Unfortunately this is the last day of the exam period. Do not make plans to leave campus before this exam, as there will be no option to take the exam early.

Problem Sets:

• Problem Set #1, due Thursday Sept 10, is to do all 10 exercises at the end of Chapter 0 of the book (available here). Here are solutions to the first problem set.
• Problem Set #2, due Thursday Sept 17. Solutions.
• Problem Set #3, due Thursday Sept 24. Solutions. To check your computations here is a continued fraction calculator.
• Problem Set #4, due Thursday Oct 1. Solutions.
• Problem Set #5, due Thursday Oct 8. Solutions.
• Problem Set #6, due Tuesday Oct 20. Solutions.
• Problem Set #7, due Tuesday Oct 27.
• Problem Set #8, due Thursday Nov 5. Solutions.
• Problem Set #9, due Friday Nov 13. Note: The original version of problem 5 was stated somewhat ambiguously and had a typo besides, so here is a revised version. The grading of this problem will take the ambiguity into account.
• Problem Set #10, due Tuesday Nov 24. If you have left town by then, you can scan your homework and send it to me by email as a pdf file (or gif, jpg, ...). The first few pages of Chapter 3 have been posted now, which should be enough to do this problem set if you missed class. I will have Office Hours Monday afternoon from 3 to 5.

Grading: Most of your grade in the course will be based on exams, the final exam counting 50% of the grade and a prelim counting 35%. The remaining 15% will be based on problem sets that will be assigned every week or two.

About this course: This is one of the standard courses that math majors can use for their algebra requirement, but a few non-majors take it too. This semester the course will be taught a little differently from usual, in that we will use pictures and geometry to illuminate algebraic concepts and constructions whenever possible.

Here is a tentative list of topics, but other things may be added later as time permits:

• Chapter 0. Pythagorean Triples. As a warm-up for the main topics in the course, we will look at integer solutions of the equation x^2 + y^2 = z^2, where there is a nice geometric method for finding formulas giving all the solutions.
• Chapter 1. The Farey Diagram. This diagram, a miniature version of which is shown at the right, gives a very nice way to picture things like the Euclidean algorithm and continued fractions.
• Chapter 2. Quadratic Forms. Here we study quadratic forms ax^2 + bxy + cy^2 over the integers, following the approach of John Conway which uses the Farey diagram to get a very beautiful and explicit picture of all the values of the form, showing not just its qualitative features but also giving a method for computing algorithmically.
• Chapter 3. Quadratic Fields. Here the topic is extensions of the rational numbers obtained by adjoining the square root of a positive or negative integer. This sheds much light on the previous topic of quadratic forms, and vice versa. It would take another semester to really get into the full theory of quadratic fields, so this will be only an introduction, focusing on a few special cases such as the Gaussian integers.