Number Theory II: Class Field Theory (Winter 23/24)

Lecture Mo 9 - 11, RUD 25, Room 1.115
Thu 9 - 11, RUD 25, Room 1.115
Problems Mo 11 - 13, RUD 25, Room 1.115

Class field theory, the study of abelian extensions of global and local fields, is one of the major achievements of mathematics in the 20th century and involves the work of many famous mathematicians such as Kronecker, Weber, Hilbert, Takagi, Artin, Hasse and Chevalley; the search for its nonabelian generalizations has led to the Langlands program which remains a challenging research area today. The starting case for the development of class field theory was the theorem of Kronecker-Weber: Every finite abelian extension \( K/\mathbb{Q} \) is contained in a cyclotomic extension, i.e. \( K\subset \mathbb{Q}(\zeta_n) \) for some \( n \). Class field theory generalizes this to abelian extensions \( K/k \) of arbitrary number fields --- it gives a complete description of all such extensions and their arithmetic in terms intrinsic to \( k \).

The course will start with a motivated introduction to the main statements of class field theory in various versions (global, local and adelic). After a brief discussion of group cohomology, we will then turn to the proof of local and global class field theory following an abstract cohomological approach by Neukirch. As prerequisites we only need basic knowledge from Algebraic Number Theory: Rings of integers in number fields, ideals and class groups, ramification of primes, local fields etc.

If you want to follow the course, please register on [moodle] (the inscription key is the last name of the Japanese mathematician who proved the existence theorem in class field theory).

Selected Literature:

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