Sharif Digital Repository

Class field theory studies abelian extensions of local and global fields. Classifying these abelian extensions was part of Hilbert’s 12th problem, which as of today is still open and is not completely solved, but that has been considerable progress towards solving it.One of these progresses is class field theory. In this theory, instead of concretely building theses extensions, we express their major properties, in terms of the arithmetic of the base field alone, which is encoded in a group called, the Idele class group (in global case) and the multiplicative group of the base field (in local case).In this thesis, we explain local and global class field theory, their relationship and the...