Brauer Group of a Field

Abstract

In this paper we discuss the Brauer group of a field and its connections with cohomology groups. Definitions involving central simple algebras lead to a discussion of splitting fields, which are the important step in the connection of the Brauer group with cohomology groups. Finally, once the connection between the Brauer group and cohomology groups is established, specific examples of cocycles associated to central simple algebras are calculated 1. Central Simple Algebras and Splitting Fields Elements of the Brauer group are equivalence classes of central simple algebras. As such, it is important to have an understanding of these algebras to understand the Brauer group. This section aims to lay the foundation for the rest of our discussion; this foundation starts with the definition of a CSA. Definitions 1.1: Fix some field k. 1. An algebra over k is a ring A along with an embedding ψ : k ↪→ A, where 1 in k maps to 1 in A. This embedding induces a scalar product that allows A to have a vector space structure over k. The image ψ(k) is often denoted k · 1, or simply k. We require ψ(k) to commute with every element of the algebra, so that the “left scalar product” and the “right scalar product” will be the same. 2. The center of an algebra A is the set Z(A) = {z ∈ A : az = za ∀a ∈ A}. If this set is the subspace k · 1, A is said to be central. 3. A (two-sided) ideal of an algebra is a (two-sided) ideal of the algebra viewed as a ring. Note that (x ·1)i = x · i ∈ I for every x ∈ k, so it has the addition stipulation of being closed under scalar multiplication. If the only ideals of A are {0} and A, A is said to be simple. 4. An algebra is a central simple algebra if it is both central and simple. Central simple algebras are often referred to as CSAs, an abbreviation we will use often. In addition to these definitions, it is useful to know when an algebra is finite-dimensional. An algebra is finite-dimensional when it is finite-dimensional as a vector space over k. All algebras will be assumed to be finite-dimensional unless stated otherwise. Examples 1.2: Common examples of CSAs include: 1. Mn(k) is a CSA over k for all n > 0. The matrix ring is equipped with a scalar product operation that multiplies each entry by an element of k. 2. Central division algebras, including k itself, are CSAs over k as well. 3. The quaternion algebra ( a,b k ) , generated by i and j with i2 = a, j2 = b, ij = −ji is a central simple algebra over k. In fact, it is either a division algebra or it is isomorphic to M2(k). A discussion of these algebras and their properties is found in [Lam].