Quaternion Numbers

This is an example of a dependent field. The Quaternion numbers can be over any field such as the Natural numbers, the integers, or the rationals or other Quaternion numbers. Quaternions are the field extension \(\mathbb{H}[\mathbb{F}] = \mathbb{F}(I,J,K)/\langle I^2=J^2=K^2=IJK=-1\rangle\). Where \(IJ=-JI\) To create a Quaternion number we can write: We can multply two Quaternion numbers together:

Hurwitz Quaternions

This is the field \( \mathbb{H}[\mathbb{Z}] \cup \mathbb{H}[\mathbb{Z}+\tfrac{1}{2}] \) $$\sqrt{1}$$