From any ordered field $\mathcal{F}$, we may construct another field $K$ whose elements consist of ordered pairs of elements of $\mathcal{F}$, and whose addition and multiplication are defined by:
- $(a, b) + (c, d) = (a + c, b + d)$
- $(a, b) \cdot (c, d) = (ac - bd, ad + bc)$
This is basically how you go from the reals to the complex numbers.
Is there a name for this process?
If $\mathcal F$ is any field where $-1$ is not a square, and you identify $\mathcal F$ with $\mathcal F\times\{0\}\subset K$, note that $(0,1)\in K$ is precisely a square root of $-1$.
I'd say that $K=\mathcal F[i]$ or $\mathcal F[\sqrt{-1}]$.