Class field theory WS 2021/22, Christopher Deninger, Maria Lünnemann

Class field theory classifies the abelian extensions L/K of a local or global number field K in terms of data determined by K. In addition, class field theory describes the decomposition behaviour of the prime ideals of o_K after extension to o_L. Understanding the non-abelian extensions of a number field is a very active topic in research - the Langlands program - and class field theory is a vital prerequisite. There are various approaches to the proofs of the main results of class field theory. For example, analytical via the theory of L-functions or cohomological by the method of Artin and Tate. In the seminar we will discuss the elegant approach by Neukirch who uses a simple field theoretic construction to turn any element in the absolute Galois group into a ''Frobenius-automorphism'' for which the reciprocity map can be written down explicitely.

Class field theory classifies the abelian extensions L/K of a local or global number field K in terms of data determined by K. In addition, class field theory describes the decomposition behaviour of the prime ideals of oK after extension to oL. Understanding the non-abelian extensions of a number field is a very active topic in research - the Langlands program - and class field theory is a vital prerequisite. There are various approaches to the proofs of the main results of class field theory. For example, analytical via the theory of L-functions or cohomological by the method of Artin and Tate. In the seminar we will discuss the elegant approach by Neukirch who uses a simple field theoretic construction to turn any element in the absolute Galois group into a ''Frobenius-automorphism'' for which the reciprocity map can be written down explicitely.

Kurs im HIS-LSF