Octonion Numbers

This is an example of a dependent field. The Octonion numbers can be over any field such as the Natural numbers, the integers, or the rationals or other Octonion numbers. Octonions are the field extension \(\mathbb{F}[e_{0..6}:(e_n)^2=-1\land e_n e_{n+1} = e_{(n+3)_7}\land n \neq m \implies e_n e_m = - e_m e_n ]\) To create a Octonion number we can write: We can multply two Octonion numbers together: $$\sqrt{1}$$