18.781 Theory of Numbers Fall Semester, 2005

Class meetings: Monday, Wednesday, and Friday 2:00–3:00, in 2-102.

Text: John Stillwell, Elements of Number Theory. You should try to read the text before class as well as after. Both your own understanding and your of catching the lecturer in a faux pas will be greatly increased. I mentioned Hardy and Wright’s Introduction to the Theory of Numbers as a recommended text not because it is a good source for the material on the syllabus, but because it is a wonderful source for a huge amount of additional mathematics.

Lecturer: David Vogan, 2-281. Telephone: 617-253-4991. E-mail: dav@math.mit.edu. My office hours are Wednesday 3–4, Thursday 3–4, or by appointment. (But in practice I’ll be in my office most of the time 9–5 weekdays, and dropping in is fine.)

Homework will be assigned in most classes. Problems assigned during each week will be collected at the beginning of the first class of the following week. You are free to consult your friends and any other sources while working on the problems, but you should write up your solutions entirely on your own. This is a place to show your understanding without time pressure.

Exams: There will be two exams during the lecture hour, on October 7 and November 9. There will be a three-hour final exam. The exams will all be closed book.

Grading: Each hour exam will be worth 100 points, the final exam will be worth 150 points, and the problem sets will be worth a total of 150 points.

Schedule

Wed 9/7  Lecture 1 1.1–1.5  Division with remainder
Fri 9/9  Lecture 2 1.6–1.9  Diophantine equations
Mon 9/12 Lecture 3 2.1–2.3  The Euclidean algorithm
Wed 9/14 Lecture 4 2.4–2.6  Prime factorization
Fri 9/16 Lecture 5 2.7–2.9 The map of relatively prime pairs
Mon 9/19 Holiday
Wed 9/21 Lecture 6 3.1-3.3  Modular arithmetic
Fri 9/23 Lecture 7 3.4–3.6 Fermat’s little theorem
Mon 9/26 Lecture 8 3.7–3.10 Repeating decimals
Wed 9/28 Lecture 9 4.1–4.7  RSA
Fri 9/30 Lecture 10 Linear algebra mod p
Mon 10/3 Lecture 11  Continued fractions and 1 + \sqrt{2}
Wed 10/5 Lecture 12  Review
Fri 10/7 Lecture 13 Exam 1 on Chapters 1–4
Mon 10/10 Holiday
Wed 10/12 Lecture 14 5.1–5.4  Pell’s equation
Fri 10/14 Lecture 15 .5.–5.6  Dirichlet’s pigeonhole principle
Mon 10/17 Lecture 16 5.7–5.9  John Conway’s river
Wed 10/19 Lecture 17 6.1–6.4  Gaussian integers
Fri 10/21 Lecture 18 6.5–6.8  Sums of two squares
Mon 10/24 Lecture 19 7.1–7.3  Quadratic integers
Wed 10/26 Lecture 20 7.4–7.6  Unique factorization?
Fri 10/28 Lecture 21 7.7–7.8  Fermat’s last theorem for n = 3

Mon 10/31 Lecture 22 8.1–8.3  Quaternions
Wed 11/2 Lecture 23 8.4–8.6  Hurwitz integers
Fri 11/4 Lecture 24 8.7–8.9 Sums of four squares
Mon 11/7 Lecture 25  Review
Wed 11/9 Lecture 26  Exam 2 on Chapters 1–8
Fri 11/11 Holiday
Mon 11/14 Lecture 27 9.1–9.2  Quadratic reciprocity
Wed 11/16 Lecture 28 9.3–9.5  Integer square roots of two
Fri 11/18 Lecture 29 9.6–9.7  Chinese remainder theorem
Mon 11/21 Lecture 30 9.8–9.9  Proof of quadratic reciprocity
Wed 11/23 Lecture 31  Hundreds of other proofs
Fri 11/25 Holiday
Mon 11/28 Lecture 32 10.1–10.3 Rings
Wed 11/30 Lecture 33 10.4–10.6 Algebraic integers
Fri 12/2  Lecture 34 11.1–11.3 Ideals and divisibility
Mon 12/5 Lecture 35 11.4–11.6 Non-principal ideals
Wed 12/7 Lecture 36 11.7–11.9 What is ideal factorization?
Fri 12/9  Lecture 37 12.1–12.3 Prime ideals
Mon 12/12 Lecture 38 12.4–12.6 Ideal factorization works
Wed 12/14 Lecture 39 12.7–12.9 Ideal classes
week of 12/15–12/19  Final Exam