Do the octonions form a field?

Question

The octonions are a noncommutative nonassociative normed division algebra over $\mathbb{R}$. Multiplication distributes over addition. Somehow, the existence of a norm implies the existence of multiplicative inverses.

Since multiplication is not associative, the nonzero octonions are not a group, but a loop. I wonder if the octonions can still be regarded as a "field", though, or if there's a more proper name for them.

Answer 1

You have answered your own question since a field must have an associative product, yet the octonions are not associative.

Although multiplication is not associative, it is alternative and flexible.

Suppose $x$ and $y$ are elements of the octionions $\mathbb{O}$.

Then

1. $(xx)y=x(xy)$
2. $x(yy)=(xy)y$
3. $(xy)x=x(yx)$

The first two properties make $\mathbb{O}$ alternative the third makes it flexible.