Why do division algebras always have a number of dimensions which is a power of $2$? Why do number systems always have a number of dimensions which is a power of $2$?


*

*Real numbers: $2^0 = 1$ dimension.

*Complex numbers: $2^1 = 2$ dimensions.

*Quaternions: $2^2 = 4$ dimensions.

*Octonions: $2^3 = 8$ dimensions.

*Sedenions: $2^4 = 16$ dimensions.


They don't. Here is a 9-dimensional associative non-commutative division algebra (over $\Bbb{Q}$):
$$
D=\left\{\left(\begin{array}{ccc}
x_1&\sigma(x_2)&\sigma^2(x_3)\\
2x_3&\sigma(x_1)&\sigma^2(x_2)\\
2x_2&2\sigma(x_3)&\sigma^2(x_1)
\end{array}\right)\bigg\vert\ x_1,x_2,x_3\in E\right\},
$$
where $E=\Bbb{Q}(\cos2\pi/7)$ and $\sigma$ is the automorphism defined by $\sigma(\cos2\pi/7)=\cos4\pi/7$.
Only over the reals are we so constrained. Topology makes a huge difference. Or, more precisely, the fact that odd degree polynomials with real coefficients always have a real zero.
