Math 514a Algebraic Number Theory Fall 1995

Final Project: You are to choose one of the topics below and write a short paper (5-10 pages) explaining the main points, to an audience at the level of the other members of the class. Due Date: Monday, December 18, 1 pm.

a) Constructing the fundamental unit in a real quadratic field via continued fractions. (Refs: 3)

b) The connection between quadratic forms and quadratic fields. Genus theory. (Refs: 1, 2, 3)

c) Local-Global principle for quadratic forms. (Refs: 3, 8)

d) Values of ζ- and L-functions at negative integers (Refs: 5, 9)

e) Fermat’s last theorem for regular primes. (Refs: 1, 9)

f) Gauss and Jacobi sums and the Hasse-Davenport relation. (Refs: 6)

g) Ramification groups and the different. (Refs: Marcus, Exercises of Chapter 4)

h) Kronecker-Weber theorem. (Refs: 9, Marcus, Exercises of Chapter 4)

i) The prime number theorem. (Refs: 4)

j) Herbrand’s criterion for regular primes. (Refs: 1, 9)

References:

1. Borevitch and Shafarevitch, Number theory

2. Cohn, A classical introduction to algebraic numbers and class fields

3. Davenport, The higher arithmetic

4. Davenport, Multiplicative number theory

5. Hida, Elementary theory of L-functions and Eisenstein series

6. Ireland and Rosen, A classical introduction to modern number theory

7. Lang, Algebraic number theory

8. Serre, A course in arithmetic

9. Washington, Cyclotomic fields