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.

Solution 1:

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.

Solution 2:

The particular family of algebras you are talking about has dimension over $\Bbb R$ a power of $2$ by construction: the Cayley-Dickson construction to be precise.