An algebra over a field such that for any elements and the equations , are solvable in . An associative division algebra, considered as a ring, is a skew-field, its centre is a field, and . If , the division algebra is called a central division algebra. Finite-dimensional central associative division algebras over may be identified, up to an isomorphism, with the elements of the Brauer group of the field . Let denote the dimension of over . If and if is the maximal subfield in ( ), then . According to the Frobenius theorem, all associative finite-dimensional division algebras over the field of real numbers are exhausted by itself, the field of complex numbers, and the quaternion algebra. For this reason the group 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 is 8. If is a finite-dimensional (not necessarily associative) division algebra over , then has one of the values 1, 2, 4, or 8.