Number of injective homomorphisms from complex numbers to quaternions

Question

We are asked to show that there are infinitely many injective ring homomorphisms from the complex numbers to the quaternions $\mathbb{H} =\mathbb{R} + \mathbb{R}i +\mathbb{R}j +\mathbb{R}k$ but I have no idea how to go about doing this (at all). I have looked at how to count the number of homomorphisms from one set to another in other examples but even those I can't get by with. Any help is much appreciated.

Answer 1

There are three obvious maps: $i \mapsto i, i \mapsto j, i \mapsto k$. The trick is that you can make new imaginary units taking a linear combination of any two.

For instance, let $\beta = (\cos \theta) i + (\sin \theta) j $ then $\beta^2 = -1$. Since $$ \left[(\cos \theta) i + (\sin \theta) j \right] \left[(\cos \theta) i + (\sin \theta) j \right] = \cos^2 \theta (i^2)+ \sin^2 \theta (j^2) = -\cos^2 \theta -\sin^2 \theta = -1. $$ Notice $ij = -ji$ so the cross-terms cancel. Now, you can map $\mathbb{C}$ to the subset of the quaternions spanned by $1, \beta$ by linearly extending the rule $i \mapsto \beta$. Since $\beta$ depends on the real parameter $\theta$ there are infinitely many such maps.