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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.

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    $\begingroup$ A "non-associative division algebra" maybe? $\endgroup$ – Lord Shark the Unknown Apr 27 '17 at 4:52
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    $\begingroup$ Fields are basically always defined to be associative and almost always defined to be commutative. But, paradoxically, Lord Shark the Unknown's comment is true that it is less rare for "division algebras" to omit associativity from the list of axioms. $\endgroup$ – rschwieb Apr 27 '17 at 14:56
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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.

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  • $\begingroup$ The usual definition of division ring also requires associativity. It's the Quaternions that are notable for being a division ring. $\endgroup$ – Hurkyl Apr 27 '17 at 6:22
  • $\begingroup$ @Hurkyl, You are correct, I will amend my answer. $\endgroup$ – John Wayland Bales Apr 27 '17 at 14:49

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