Algebraic Number Theory
Mark Bashmakov

1. Valuations of rationals. Ostrowski’s Theorem. [B-Sh]

2. Local methods
2.1. Hensels Lemma.
2.2. Structure of the field of p-adic numbers.
2.3. Quadratic forms over p-adic field.
2.4. Computing Hilbert symbol.
2.5. Hilbert Reciprocity Law.
2.6. Extension of the field Qp. Inertia degree and Ramification index.

3. Theory of divisors.
Dedeking Theorem. [A, B-Sh]

4. Geometrical Methods. [B-Sh, R, S-T]
4.1. Minkowski’s Lemma.
4.2. Class group.
4.3. Dirichlet’s Unit Theorem.

5. Main examples. [R, A, S-T]
5.1.Quadratic fields.
5.2. Cyclotomic fields.

6. Decomposition of integers.
6.1. Extension of valuations.
6.2. Factoring primes.
6.3. Discriminant.

7. Analytical Methods. [B-Sh, R]
7.1. Dedekind S-function.
7.2. Formulas for class-number.
7.3. Dirichlet Theorem on primes in arithmetical progression.

8. Galois Theory.[K]
9. Class field Theory. [K]
10. Examples. Cubic fields.

Literature
1. [B-Sh] Z. Borevich, I. Shafarevich, Number Theory, Academic Press, 1966.
2. [A] Alaca & Williams, Introductory Algebraic Number Theory, Cambridge Un. Press, 2004.
3. [R] P. Rebenboim, Classical Theory of Algebraic Numbers, Springer, 2001.
4. [K] H. Koch, Algebraic Number Theory, Sprunger, 1997.
5. [S] J.-P. Serre, Course in Arithmetic, Springer, 1996.
6. [G] F. Gouvea, p-adic Numbers, Springer, 1997.
7. [S-T] Stewert & Tall, Algebraic Number Theory, Chapman & Hall, 1979.