# central simple algebra

Let $K$ be a field. A central simple algebra $A$ (over $K$) is an algebra $A$ over $K$, which is finite dimensional as a vector space over $K$, such that

• $A$ has an identity element, as a ring

• $A$ is central: the center of $A$ equals $K$ (for all $z\in A$, we have $z\cdot a=a\cdot z$ for all $a\in A$ if and only if $z\in K$)

• $A$ is simple: for any two sided ideal $I$ of $A$, either $I=\{0\}$ or $I=A$

By a theorem of Brauer, for every central simple algebra $A$ over $K$, there exists a unique (up to isomorphism) division ring $D$ containing $K$ and a unique natural number $n$ such that $A$ is isomorphic to the ring of $n\times n$ matrices with coefficients in $D$.

Title central simple algebra CentralSimpleAlgebra 2013-03-22 11:49:08 2013-03-22 11:49:08 djao (24) djao (24) 7 djao (24) Definition msc 16D60 msc 70K75