MA 356: Class field theory: a course in arithmetic geometry

Credits: 3:0

Pre-requisites :

1. Basic algebra, commutative rings, Noetherian rings, basic number theory,
2. basic knowledge of Galois theory of fields.
3. Number fields and finite fields.

Syllabus:

1. Review of Dedekind domains and rings of integers in number fields.
2. Topology of discrete valuation fields.
3. Group cohomology and Galois cohomology.
4. Brauer group.
5. Brauer group of local fields.
6. Hasse principle for Brauer group.
7. Norm subgroups and their openness.
8. Class field theory for local and global fields.
9. Class field theory for compact curves over finite fields.
10. Class field theory for open curves over finite fields.

Suggested books and references:

1. J. P. Serre, Local fields, Graduate Texts in Mathematics. Vol. 67. Springer-Verlag. New York-Heidelberg. 1979.
2. J. Milne, Lecture notes on class field theory

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