Mo 13:40 - 15:30, SAZ 02 We 15:40 - 17:30, SAZ 02

- 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.

- We 13.09.06 problems
- We 02.10.06 problems; solutions
- Mo 16.10.06 problems;
in addition,
**please**read Barry Mazur's article on algebraic numbers. - Mo 23.10.06 problems; solutions
- Mo 04.11.06 problems; solutions

- Mo 11.09.06: Euler and the failure of unique factorization
- We 13.09.06: Euclidean rings; Principal Ideal Domains
- Mo 18.09.06: Unique Factorization Domains; Ideals
- We 20.09.06: Ideals
- Mo 25.09.06: Algebraic Integers, modules
- We 27.09.06,
**4:40 - 5:30 pm**Ideals. - Mo 02.10.06: Unique Factorization into prime ideals
- We 04.10.06
**3:40 - 4:30**examples - Mo 09.10.06: Units
- We 11.10.06
**3:40 - 5:30**: Units - Mo 16.10.06 Class groups
- We 18.10.06: NO CLASS, moved to 11.10.06
- Mo 23.10.06 no class
- We 25.10.06 no class
- Mo 30.10.06 Class group calculations continued
- We 02.11.06 Diophantine equations
- Mo 06.11.06 Genus theory and quadratic reciprocity
- We 09.11.06
**3:40 - 5:30**: quadratic forms - Mo 13.11.06 quadratic forms
- We 15.11.06 no class; moved to 09.11.
- Sa 18.11.06
**12:00 - 14:00, SAZ04: Midterm 1**covering everything up to class group calculations. Proctor: Adem Ersin Ureyen. Here are the solutions - Mo 20.11.06 reduction of quadratic forms
- We 22.11.06 quadratic forms
- Mo 27.11.06 Bhargava composition of quadratic forms
- We 29.11.06 No Class.
- Mo 04.12.06 Class groups in orders
- We 06.12.06 Bijection between forms and modules.
- Mo 11.12.06 Quadratic Forms in Function Fields
- We 13.12.06 Elliptic and Hyperelliptic Curves. Here are the updated lecture notes
- Sa 16.12.06 15:00, SAZ 04, Midterm 2. Here are the solutions. Average 76/100
- Mo 18.12.06 Elliptic and Hyperelliptic Curves
- We 20.12.06 No class (16.12.)
- Jan 04, 2007
**NO FINAL**(postponed) - Jan 06, 2007; 14:00 - 16:00, SAZ03 final.

- 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.