Number fields

Algebraic integers, ideal factorization, ramification, geometry of numbers, units, class groups, quadratic and cyclotomic fields, and zeta and L-functions.

The course has seventeen lessons and includes reproducible finite certificates. Author checks do not imply independent review of the entire course or every earlier dependency.

1. Algebraic integers and rings of integers
2. Discriminants and integral bases
3. Discrete valuation rings and Dedekind domains
4. Norms of ideals, the ideal class group, and modules over Dedekind domains
5. Decomposition of primes in extensions
6. Hilbert's ramification theory in Galois extensions
7. Lattices, Minkowski's theorem and the Minkowski embedding
8. Finiteness of the class number
9. Dirichlet's unit theorem
10. Quadratic fields: ideal classes and binary quadratic forms
11. Orders in number fields and their Picard groups
12. Cyclotomic fields
13. Units and class numbers of cyclotomic fields; Kummer's theorem for regular primes
14. The different and the discriminant
15. Counting ideals of bounded norm
16. The Dedekind zeta function and the analytic class number formula
17. Abelian number fields and Dirichlet L-functions at s = 1