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.
Neukirch, Jürgen: Class field theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 280. Springer-Verlag, Berlin, 1986. viii+140 pp. ISBN: 3-540-15251-2
Neukirch, Jürgen: Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher. With a foreword by G. Harder. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322. Springer-Verlag, Berlin, 1999. xviii+571 pp. ISBN: 3-540-65399-6