information:classification, quality:correction
Center

The center of a noncommutative ring is the subring of elements c such that cx = xc for every x. The center of the quaternion algebra is the subfield of real quaternions. In fact, it is a part of the definition that the real quaternions belong to the center. Conversely, if q = a + b i + c j + d k belongs to the center, then

0 = i q − q i = 2 c i j + 2 d i k = 2 c k − 2 d j, {\displaystyle 0=\mathbf {i} \,q-q\,\mathbf {i} =2c\,\mathbf {ij} +2d\,\mathbf {ik} =2c\,\mathbf {k} -2d\,\mathbf {j} \,,}

and c = d = 0. A similar computation with j instead of i shows that one has also b = 0. Thus q = a is a real quaternion.

The quaternions form a division algebra. This means that the non-commutativity of multiplication is the only property that makes quaternions different from a field. This non-commutativity has some unexpected consequences, among them that a polynomial equation over the quaternions can have more distinct solutions than the degree of the polynomial. For example, the equation z + 1 = 0, has infinitely many quaternion solutions, which are the quaternions z = b i + c j + d k such that b + c + d = 1. Thus these "roots of –1" form a unit sphere in the three-dimensional space of vector quaternions.