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The popular story of the discovery of the quaternions goes very roughly as follows.

William Rowan Hamilton has interested in the construction of an algebra of triplets that would in some ways be analogous to the complex numbers. He eventually realized this can not be done, but that one can construct such an algebra for $4$-tuples thereby creating the quaternions.

My questions are:

  1. What are the properties that Hamilton desired this algebra of triplets would have?
  2. Why was or would it be desirable to have such an algebra?
  3. Why can't it be done?

I'd appreciate any sources or answers to these questions.

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Dropping the requirement of commutativity, Hopf generalized his result: Any finite-dimensional real division algebra must have dimension a power of 2. – Will Jagy Nov 11 '13 at 21:58

As far as "why can't it be done" goes, there are several potential interpretations.

Some links and discussions of proofs can be found in this thread. These primarily focus on Clifford algebras or some results of Frobenius, and only cover the associative case. The non-associative case is also known, and is covered in the next paragraph.

There is an alternative explanation that isn't mentioned that thread, which was established in Adams, J. F. (1962). "Vector Fields on Spheres". Annals of Mathematics 75: 603–632. Zbl 0112.38102. This concerns establishing the maximum number of linearly independent smooth (or continuous, even) vector fields over spheres of a given (finite) dimension. It turns out that having a well-defined multiplication on $\Bbb R^n$ with inverses and no zero divisors is equivalent to having the largest possible number of such linearly independent vector fields available. And his results show that such a multiplication exists only on $\Bbb R, \Bbb R^2, \Bbb R^4,$ and $\Bbb R^8$. The first three are the real numbers, complex numbers, and quaternions. The last one is a non-associative division algebra over the reals known as the octonions.

For $\Bbb R^3$ in particular, this result is sometimes stated as "you can't comb the hair on a coconut", or The Hairy Ball Theorem. Any attempt to comb the hairs continuously will result in a "cowlick"—a discontinuity, namely. The hairs can be thought of as representing the vector fields in the Adams interpretation.

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