Class Field Theory

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Motevassel, Mohyeddin | 2016

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 48493 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Jafari, Amir
Abstract:
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 proofs of their major theorems. In the process, we explain analytic and algebraic number theory, reciprocity laws, central simple algebras, duality in locally compact groups, etc
Keywords:
BRAUER GROUP ; Frobenius Group ; Cyclic Algebra ; DUAL GROUP ; Abelian Extension ; Finite and Infinite Primes ; Idele Class Group ; Artin Map ; Hilbert Class Field ; Locally Compact Groups