TITLE: Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?
QUESTION [15 upvotes]: I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the complex field, where we can see a number as a $2$ dimensional vector.
In this paper, he explains why is impossible to create a field with $3$ components, that is an extension of the complex field (in other words, it respects addition, and multiplication in the same way...).
Here it is:

![enter image description here][1]
As you can see, it goes through all the possibilities and proves that it is impossible.
**The paper, however, does not explain why $j^2=-1$. It could be anything! Why $-1$?** The article itself is pretty intuitive, but this aspect kills me.

Later, in the article, it says that we should instead consider a 4th component called $k$, such that $k^2=-1$ (also, $i$ and $j$ too). 
[Here is the paper][2]

**EDIT**: [this][3] paper by Rupert Shuttleworth turned out to be extremely helpful (mirrored [here][4] on archive.org)

REPLY [4 votes]: I don't know if this is what Hamilton had in mind, but: suppose you've decided that what you want to do is to write down a division algebra $D$ over $\mathbb{R}$ bigger than the complex numbers $\mathbb{C}$. Any element $d \in D$ of $D$ generates a subalgebra $\mathbb{R}[d]$ of $D$ which is a commutative division algebra (in other words, a field) over $\mathbb{R}$ and hence (because $\mathbb{C}$ is algebraically closed) must be isomorphic to either $\mathbb{R}$ or $\mathbb{C}$. 
If $d \not \in \mathbb{R}$, then this subalgebra $\mathbb{R}[d]$ must be isomorphic to $\mathbb{C}$. Any such isomorphism realizes $d$ as a complex number $a + bi, b \neq 0$, and hence up to a real change of coordinates we might as well have chosen $d$ to correspond to $i$ under this isomorphism; that is, to satisfy $d^2 = -1$. With a bit more work this argument shows that $D$ is generated by elements that square to $-1$.