TU 13:40 - 15:30, SAZ 03 TH 13:40 - 14:30, SAZ 20

- I. Quadratic Number Fields. Here we will discuss
failure of unique factorization, ideal arithmetic,
class number computation, and units. Applications:
Bachet-Mordell equations y
^{2}= x^{3}+k, some simple 3-descents on elliptic curves like x^{3}+y^{3}= Az^{3}, possibly the cubic reciprocity law. - II. Quadratic Function Fields. These are quadratic
extensions of the rational function fields F
_{p}[X]; their arithmetic has a lot in common with quadratic number fields. - III. Arithmetic of Elliptic and Hyperelliptic Curves. These can essentially be identified with quadratic extensions of function fields.

- Th 15.09.05 problems, solutions
- Tu 04.10.05 problems, solutions
- Tu 11.10.05 problems, solutions
- Tu 08.11.05 problems, solutions
- Th 01.12.05 problems, solutions

- Tu 13.09.05: Euler and the failure of unique factorization
- Th 15.09.05: Algebraic Integers (ring of integers; basic properties of ideals)
- Tu 20.09.05: Ideals and modules
- Th 22.09.05: NO CLASS (moved to Th. 29.09.05)
- Tu 27.09.05: Unique Factorization into prime ideals
- Th 29.09.05: Decomposition law in quadratic number fields.
- Tu 04.10.05: The Pell equation
- Th 06.10.05: Some diophantine equations
- Tu 11.10.05: Principal ideal tests, Computing units. This is the article by Lenstra that I recommended.
- Th 13.10.05: NO CLASS (moved to Th. 27.10)
- Tu 18.10.05: Factoring integers
- Th 20.10.05: Class groups. Here are Chapters 1-6.
- Tu 25.10.05: Quadratic Reciprocity. Here is a review sheet.
- Th 27.10.05: Midterm 1 (solutions)
- Tu 01.11.05: discussion of midterm
- Th 03.11.05: No class
- Tu 08.11.05: Applications to cryptography; here are Chapters 1-6 with an expanded Chapter 6.
- Th 10.11.05: Applications to cryptography
- Tu 15.11.05: Applications to cryptography
- Th 17.11.05: Binary Quadratic Forms.
- Tu 22.11.05: Reduction
- Th 24.11.05: No class (moved to Fr. 16.12)
- Tu 29.11.05: Here are a few practice problems
- Th 01.12.05: Reduction II
- Tu 06.12.05: Gauss Composition
- Th 08.12.05: Gauss Composition II
- Tu 13.12.05: Midterm 2; average 38.3/100. Here are the solutions.
- Th 15.12.05: discussion of midterm, isomorphism
- Fr 16.12.05, 11:00 - 12:00; practice hour
- Mo 19.12.05, 1:30; algebraic number theory; practice for final.
- We 21.12.05, 9:30 - 11:30, SAZ04 Final.
Problems
Solutions.
Average: 53
- Final average of students who turned in homework regularly: 65
- Final average of students who turned in homework irregularly: 42
- Final average of students who did not turn in homework: 27

- The lecture notes (still details missing in some proofs concerning the group structure on classes of quadratic forms . . .)

- H. Cohn, Advanced Number Theory. One of the few books with a readable account of quadratic forms.
- D. Flath, Introduction to number theory. Probably the best presentation of the theory of quadratic forms.
- Cox, Primes of the form x
^{2}+ ny^{2}: Chapter 1 contains the theory of binary quadratic forms; the other two chapters deal with class field theory and complex multiplication. - Marcus: Number Fields. My favorite for the theory of general number
fields; it has
*lots*of exercises. - Ireland and Rosen: A classical introduction to modern number theory. An all-time classic.
- Murty and Esmonde: another problem collection.
- Swinnerton-Dyer: A Brief Guide to Algebraic Number Theory: concise, not expensive; if you want to buy a book, this is it.