6
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Cross-posted from Math SE because I felt like it might be too obscure for there. I'm sorry if this is the wrong place for it.

EDIT: This question now has an answer over there


The finite-dimension division algebras over the reals are:

  • $\Bbb R$: the reals (dimension 1)
  • $\Bbb C$: the complex numbers (dimension 2)
  • $\Bbb H$: the quaternions (dimension 4)
  • $\Bbb O$: the octonions (dimension 8)
  • some other dimension 2 and dimension 8 things

An example of a dimension-2 division algebra other that $\Bbb C$ is $(\Bbb C,*)$ with $a*b:=\overline{ab}$ (that is, the complex conjugate of the usual multiplication). This gives you $1*1=1$, $1*i=i*1=-i$, and $i*i=-1$. You'll notice that $1*a$ does not necessarily equal $a$; that is, this algebra is not unital. There exist nonunital division algebras of dimension 8 as well.

Are there any unital division algebras of dimension 8 (other than $\Bbb O$)? Such an algebra cannot be alternative, nor can it have a norm (as each of these, together with the dimension 8 condition, uniquely define the octonions).

(EDIT: The nonunital algebra defined above has a norm, but the only unital normed algebras are $\Bbb R$, $\Bbb C$, $\Bbb H$, and $\Bbb O$.)

Strangely, I haven't been able to find a source one way or another online, which is weird because it seems like it would close up the search for real division algebras. So, does such a thing exist?

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  • $\begingroup$ This does not answer your question, but evidently there are examples over finite fields. tandfonline.com/doi/abs/10.1080/… $\endgroup$ – John Wayland Bales Sep 9 '18 at 0:41
  • $\begingroup$ It occurs to me that I should have written some other dimension 2, dimension 4, and dimension 8 things in that last bullet point since $a*b=\overline{ab}$ generalizes to the quaternions as well. $\endgroup$ – Akiva Weinberger Sep 9 '18 at 5:07
  • $\begingroup$ What about dimension 4? Is it known? $\endgroup$ – abx Sep 9 '18 at 6:18
  • $\begingroup$ @abx You know, I assumed nonassociative implies dimension 8, but I guess that might not actually be true. (Associative implies $\Bbb R$, $\Bbb C$, or $\Bbb H$, with or without the unital condition, at least according to Wikipedia.) If someone presented a unital dimension 4 division algebra other than $\Bbb H$ I'd accept it as an answer also. $\endgroup$ – Akiva Weinberger Sep 9 '18 at 6:23
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    $\begingroup$ In general, a division algebra satisfies that the equations $xz = y$ and $z'x = y$ always has a solution for any nonzero $x$. For alternative algebras this implies the existence of a unit, but in general it doesn't. $\endgroup$ – arsmath Sep 9 '18 at 11:16

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