Course Identification

1. Gaussian integers and integrality

2. Ideals

3. Lattices & Minkowski's "geometry of numbers"

4. The Class Number

5. Dirichlet's unit theorem

6. Extensions of Dedekind domains

7. Hilbert's ramification theory

8. Cyclotomic fields 

9. Localization

10. Orders

11. One-dimensional schemes

12. Function fields

13. p-adic numbers

14. Valuations & completions

15. Local fields

The students will learn the relations between algebra and number theory, and understand the origins of many algebraic notions. The students will acquire the terminology and basic notions of modern number theory.

1. J. Neukirch "Algebraic number theory"
2. S. Lang "Algebraic number theory"
3. D. Marcus "Number Fields"