Class field theory is one of the high peaks in the development of number theory in the first half of the 20th century. It aims at understanding the finite Galois extensions with abelian Galois group of a number field K (i.e. a finite extension of the rational numbers). One application is for example that the finite abelian Galois extensions of the rational numbers are exactly those extensions, which are contained a cyclotomic field extensions. In order to obtain the description for a number field, which is considered to be a global field, one first proves a local version, involving local fields. These fields arise by completing a number field along its various primes. By local class field theory the abelian Galois extensions of a local field L correspond to certain subgroups of L\{0}.
In the course we will discuss infinite Galois theory, global and local fields and give the formulation of global and local class field theory. We will give the main ideas and constructions in the proof and discuss details as time permits.
Prerequisites: Basic knowledge in (finite) Galois theory, commutative algebra and number theory.
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