Brauer group
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In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer.
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[edit] Construction of the Brauer group
A central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A, which is a simple ring, and for which the center is exactly K. For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The finite-dimensional division algebras with center R (that means the dimension over R is finite) are the real numbers and the quaternions by a theorem of Frobenius.
Given central simple algebras A and B, one can look at the their tensor product A ⊗ B as a K-algebra (see tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterisation: a central simple algebra over K is a K-algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of K.
Given this closure property for CSAs, they form a monoid under tensor product. To get a group, apply the Artin-Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n,D) for some division algebra D. If we look just at D, rather than the value of n, the monoid becomes a group. That is, if we impose an equivalence relation identifying M(m,D) with M(n,D) for all integers m and n at least 1, we get an equivalence relation; and the equivalence classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra Aop (the opposite ring with the same action by K since the image of K → A is in the center of A). In other words, for a CSA A we have A ⊗ Aop = M(n2,F), where n is the degree of A over F. (This provides a substantial reason for caring about the notion of an opposite algebra: it provides the inverse in the Brauer group.)
[edit] Examples
The Brauer group for an algebraically closed field or a finite field is the trivial group with only the identity element.
The Brauer group Br(R) of the real number field R is a cyclic group of order two: there are just two types of division algebras, R and the quaternion algebra H. The product in the Brauer group is based on the tensor product: the statement that H has order two in the group is equivalent to the existence of an isomorphism of R-algebras: H ⊗ H ≅ M(4,R), where the RHS is the ring of 4×4 real matrices.
Tsen's theorem implies that the Brauer group of a function field in one variable over an algebraically closed field vanishes.
[edit] Further theory
In the further theory, the Brauer group of a local field is computed (it turns out to be canonically isomorphic to Q/Z for any local field, of characteristic 0 or characteristic p) and the results are applied to global fields. This gives one approach to class field theory, which was the first approach that allowed global class field theory to be derived from local class field theory; historically it had been the other way around at first. It also has been applied to Diophantine equations. More precisely, the Brauer group Br(K) of a global field K is given by the exact sequence
where the direct sum in the middle is over all (archimedean and non-archimedean) completions of K and the map to 
is addition, where we interpret the Brauer group of the reals as (1/2)Z/Z. The group Q/Z on the right is really the "Brauer group" of the class formation of idele classes associated to K.
In the general theory the Brauer group is expressed by factor sets; and expressed in terms of Galois cohomology via
Here, not assuming K to be a perfect field, Ks is the separable closure. When K is perfect this is the same as an algebraic closure; otherwise the Galois group must be defined in terms of Ks/K even to make sense.
A generalisation via the theory of Azumaya algebras was introduced in algebraic geometry by Grothendieck.
