Prove the polynomial ring and algebra of quaternions isomorphisms

Question

I saw an answer on a question Prove the polynomial ring and field isomorphisms, but can somebody explain why with $x=j$ and $ix=k$, the elements $i,j,k$ satisfy all the defining relations they are supposed to for the quaternion algebra $\mathbb{H}$?

Answer 1

Normally you would leave a comment at the solution asking for clarification, but here I see you probably don't have the requisite reputation to do so, so this is OK.

You've explained your question in a comment like this:

$j\cdot i=xi=ix=k$ So, I think the answer isn't correct or I didn't understand something.

Yes, you've misunderstood something. In the solution there, $xi\neq ix$. If you read the posted question more closely, we are working with something called a twisted polynomial ring, where $x\lambda=\alpha(\lambda) x$ where $\alpha$ is a ring homomorphism. (The post says skew polynomial ring but in my mind this describes a more general construction.)

The ring homomorphism in this construction is complex conjugation, so in the twisted polynomial ring described there, $xi=\alpha(i)x=-ix$