Editing Quaternion (section)

=== Center ===
The [[center (ring theory)|''center'']] of a [[noncommutative ring]] is the subring of elements {{mvar|c}} such that {{math|1=''cx'' = ''xc''}} for every {{mvar|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 {{math|''q'' {{=}} ''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} belongs to the center, then

<math display=block>0 = \mathbf i\,q - q\,\mathbf i = 2c\,\mathbf{ij} + 2d\,\mathbf{ik} = 2c\,\mathbf k - 2d\,\mathbf j,</math>

and {{math|''c'' {{=}} ''d'' {{=}} 0}}. A similar computation with {{math|'''j'''}} instead of {{math|'''i'''}} shows that one has also {{math|''b'' {{=}} 0}}. Thus {{math|''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 {{nowrap|{{math|''z''<sup>2</sup> + 1 {{=}} 0}},}} has infinitely many quaternion solutions, which are the quaternions {{math|''z'' {{=}} ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} such that {{math|''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup> {{=}} 1}}. Thus these "roots of –1" form a [[unit sphere]] in the three-dimensional space of vector quaternions.