Brauer group

1 Algebraic view

Let $K$ be a field. The Brauer group $\mathrm{Br}(K)$ of $K$ is the set of all equivalence classes of central simple algebras over $K$, where two central simple algebras $A$ and $B$ are equivalent if there exists a division ring $D$ over $K$ and natural numbers $n,m$ such that $A$ (resp. $B$) is isomorphic to the ring of $n\times n$ (resp. $m\times m$) matrices with coefficients in $D$.

The group operation in $\mathrm{Br}(K)$ is given by tensor product: for any two central simple algebras $A,B$ over $K$, their product in $\mathrm{Br}(K)$ is the central simple algebra $A{\otimes }_{K}B$. The identity element in $\mathrm{Br}(K)$ is the class of $K$ itself, and the inverse of a central simple algebra $A$ is the opposite algebra $A^{\mathrm{opp}}$ defined by reversing the order of the multiplication operation of $A$.

2 Cohomological view

The Brauer group of $K$ is naturally isomorphic to the second Galois cohomology group $H^2(\mathrm{Gal}(K^{\mathrm{sep}}/K),{(K^{\mathrm{sep}})}^{\times })$. See http://www.math.harvard.edu/ elkies/M250.01/index.htmlhttp://www.math.harvard.edu/ elkies/M250.01/index.html Theorem 12 and succeeding remarks.

Title	Brauer group
Canonical name	BrauerGroup
Date of creation	2013-03-22 11:49:31
Last modified on	2013-03-22 11:49:31
Owner	djao (24)
Last modified by	djao (24)
Numerical id	13
Author	djao (24)
Entry type	Definition
Classification	msc 16K50
Defines	opposite algebra