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I've been reading https://github.com/GleasSpty/MATH-104-----Introduction-to-Analysis, and the author formulates the integers as the smallest (by inclusion under isomorphism) nontrivial totally ordered cring that contains the natural numbers, the rationals as the smallest totally ordered field that contains the integers, and the reals as the smallest dedekind-complete (or cauchy-complete) totally ordered field that contains the rationals. Similarly, there's the algebraic numbers which are the smallest (edit: they're not totally ordered) algebraically complete field that contains the rationals, and the complex numbers which are both algebraically complete and dedekind-complete.

Is there a similar statement for the Quaternions/Octonions?

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    $\begingroup$ Algebraically closed fields can't be totally ordered (at least, not in a way that respects the field operations). It's easy to see that $i$ and $0$ have to be incomparable. $\endgroup$
    – Robert Shore
    Commented Aug 13, 2020 at 3:22
  • $\begingroup$ Quaternions are the largest associative Euclidean Hurwitz algebra, and octonions are the largest period. $\endgroup$
    – anon
    Commented Aug 13, 2020 at 3:51
  • $\begingroup$ Agree with Robert: complex numbers are not Dedekind-complete since they cannot even be totally ordered (in a way that respects the algebraic structure). $\endgroup$
    – lisyarus
    Commented Aug 13, 2020 at 5:41
  • $\begingroup$ Another note: the order really comes into play for reals only; there is only over way to put an order on integers and reals, so specifying that e.g. integers are a totally ordered ring (as opposed to just a ring) doesn't add anything. $\endgroup$
    – lisyarus
    Commented Aug 13, 2020 at 5:49

2 Answers 2

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By Frobenius' theorem, the quaternions $\Bbb{H}$ can be characterized as the smallest noncommutative division ring that contains $\Bbb{C}$.

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    $\begingroup$ It should say "the only finite-dimensional", since there are infinite-dimensional noncommutative division rings over reals. $\endgroup$
    – lisyarus
    Commented Aug 13, 2020 at 5:39
  • $\begingroup$ Took out the "only"! $\endgroup$
    – Rivers McForge
    Commented Aug 13, 2020 at 5:42
  • $\begingroup$ I believe it should be noted that "smallest" here is in terms of dimension, not inclusion (compare to e.g. rational numbers being the smallest char-zero field wrt inclusion). $\endgroup$
    – lisyarus
    Commented Aug 13, 2020 at 5:47
  • $\begingroup$ Ok, I took that other stuff out too, just so it doesn’t get nitpicked to death any more. $\endgroup$
    – Rivers McForge
    Commented Aug 13, 2020 at 6:14
  • $\begingroup$ Well, in my opinion changing from "precisely wrong" to "precisely right" is better than to "ambiguously right", but that's your answer after all. $\endgroup$
    – lisyarus
    Commented Aug 13, 2020 at 6:33
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If I understand correctly what you are looking for, then no, I do not believe there is a comparable statement to be made about the Quaternions and Octonions.

Quaternions and Octonions are useful, but I would not say that they are remarkable because of their structure in the same way that Integers, Rationals, Reals, Algebraic Numbers, or Complex Numbers are. Quaternions are not commuatative, and Octonians are neither commutative nor associative. So rather than having more structure than any other set of numbers, they actually have less.

Quaternions are specifically useful because they can represent mathematical objects (i.e. 3D rotations) in a way which makes them easy to manipulate and reason about. It is easy to compose and interpolate rotations when they are represented as Quaternions because these operations are "encoded" in the algebra of the Quaternions.

It is my understanding that Octonions also represent some algebraic objects more elegantly than other sets of numbers do, and can be used to practical ends for that reason, but I am not familiar with how they are used.

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