The Brauer Group of Rational Numbers

TL;DR

This paper reviews the determination of the Brauer group of rational numbers, focusing on elementary methods to understand its structure through real and local fields, filling gaps left by more advanced treatments.

Contribution

It provides a detailed, elementary exposition of the steps involved in determining Br(Q), including original proofs and comprehensive explanations.

Findings

Determined Br(R), the Brauer group of real numbers.

Analyzed Br(k_v), the Brauer groups of local fields.

Constructed maps from Br(Q) to Br(R) and Br(Q_p) to understand Br(Q).

Abstract

In this project, we will study the Brauer group that was first defined by R. Brauer. The elements of the Brauer group are the equivalence classes of finite dimensional central simple algebra. Therefore understanding the structure of the Brauer group of a field is equivalent to a complete classification of finite dimensional central division algebras over the field. One of the important achievements of algebra and number theory in the last century is the determination of Br(Q), the Brauer group of rational numbers. The aim of this dissertation is to review this project, i.e., determining Br(Q). There are three main steps. The first step is to determine Br(R), the Brauer group of real numbers. The second step is to identify Br(k_\nu), the Brauer group of the local fields. The third step is to construct two maps Br(Q) to Br(R) and Br(Q) to Br(Q_p) and to use these two maps to understand Br(Q). This dissertation completed the first two steps of this enormous project. To the author's knowledge, in literature there is no document including all the details of determining Br(Q) and most of them are written from a advanced perspective that requires the knowledge of class field theory and cohomology. The goal of this document is to develop the result in a relatively elementary way. The project mainly follows the logic of the book [6], but significant amount of details are added and some proofs are originated by the author, for example, 1.2.6, 1.4.2(ii), 4.2.6, and maximality and uniqueness of 5.5.12.