# Division algebra

An algebra $A$ over a field $F$ such that for any elements $a \neq 0$ and $b$ the equations $ax = b$, $ya = b$ are solvable in $A$. An associative division algebra, considered as a ring, is a skew-field, its centre $C$ is a field, and $C \supseteq F$. If $C = F$, the division algebra $A$ is called a central division algebra. Finite-dimensional central associative division algebras over $F$ may be identified, up to an isomorphism, with the elements of the Brauer group $B( F )$ of the field $F$. Let $[ A: F ]$ denote the dimension of $A$ over $F$. If $A \in B( F )$ and if $L$ is the maximal subfield in $A$ ($L \supseteq F$), then $[ A: F ] = {[ L: F ] } ^ {2}$. According to the Frobenius theorem, all associative finite-dimensional division algebras over the field of real numbers $\mathbf R$ are exhausted by $\mathbf R$ itself, the field of complex numbers, and the quaternion algebra. For this reason the group $B( \mathbf R )$ is cyclic of order two. If the associativity requirement is dropped, there is yet another example of a division algebra over the field of real numbers: the Cayley–Dickson algebra. This algebra is alternative, and its dimension over $\mathbf R$ is 8. If $A$ is a finite-dimensional (not necessarily associative) division algebra over $\mathbf R$, then $[ A: \mathbf R ]$ has one of the values 1, 2, 4, or 8.