On the Brauer group of a local field Open Access

Joseph, Adheep (Spring 2022)

Permanent URL: https://etd.library.emory.edu/concern/etds/m613mz79m?locale=en


This thesis focuses on the study of Brauer groups via Galois Cohomology. In particular, it will cover Wedderburn theory of central simple algebras over general fields, group cohomology and Galois Cohomology, generic splitting, and a description of the Brauer group as a Galois Cohomology group. With a blend of arithmetic results from class field theory, the main aim of the thesis will be the determination of the Brauer group of a local field and establishing the reciprocity sequence for the Brauer group of a number field.

Table of Contents

Chapter 1: Central Simple Algebras and The Brauer Group

Wedderburn's Theorem Central Simple Algebras (Some Results) Skolem-Noether Theorem The Brauer Group Relative Brauer Group

Chapter 2: The Cohomological Brauer Group

Profinite Groups Cohomology of Profinite Groups Hilbert's Theorem 90 Brauer Group as Galois Cohomology Group Some Computations Central Simple Algebras Over Complete Discrete Valued Fields

Chapter 3: Applications to Class Field Theory

Cohomological Dimension Brauer Group of a Local Field Some Concluding Remarks


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