# 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 |