Algebraic Number Theory
2020/9/14 – 2021/1/18 Every Mondays 9:20-11:50
R609, Astronomy-Mathematics Building, NTU
Speaker:
Chia-Fu Yu 余家富 (Academia Sinica)
Organizer:
Chieh-Yu Chang 張介玉 (National Tsing Hua University)
Chia-Fu Yu 余家富 (Academia Sinica)
Background and Purpose：
This is an algebraic number theory course with aim to cover Class Field Theory. We first plan to discuss in details on algebraic background underlying the ring of integers and their extensions as Dekekind domains and fundamental results of them. Then we will discuss the standard topics of number fields including ramifications, valuations, adeles and ideles.
The central part is to cover Class Field Theory, we follow Lang’s book subject to assuming necessary results proved by analytic method. The remaining part is to develop cohomology of groups and Galois cohomology.
Outline
We divide the course into 4 parts
1: General theory on Dedekind domains and their extensions: trace, norm, different, discriminant, DVR, primary decomposition and structure theorem of modules.
2. Rings of integers of number fields, norm of ideals, ideal class groups, finiteness, Dirichlet’s unit theorem, ideles and adeles, cyclotomic extensions.
3. Norm indices, Hilbert’s Theorem 90, local-global principle, the Artin map, Main Theorem of Global Class Field Theory, Kronecker-Weber theorem, Kummer theory, the existence theorem.
4: Cohomology of Groups, Galois cohomology.