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Taiwan Mathematics School: Algebraic Number Theory
09:20-11:50, every Monday, September 14, 2020 - January 18, 2021
R609, Astronomy-Mathematics Building, NTU

Chia-Fu Yu (Academia Sinica)

Chieh-Yu Chang (National Tsing Hua University)


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.



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,

Contact: murphyyu@ncts.ntu.edu.tw

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