Adjoin complex numbers to an arbitrary field?

Question

This is probably nonsense but I'm throwing it out there. I don't think I can even explain the question very well: Has anyone seen bizarre things such as adjoining, say $i$ or $\pi$, to say a finite field? Would anything such as $zm$, for $z \in \mathbb{C}$ and $m \in \mathbb{F}_q$ be made to have any useful meaning? In the past the invention of things such as $\sqrt{-1}$, though intuitive enough for us these days, must have been rather abstract and "nonsensical" for folks those days. Yet complex numbers are curiously useful to solve problems about things which should have nothing to do with these.

Answer 1

There is a way to make sense of what you're asking.

If you want to a adjoin a single algebraic number to $\mathbb{Q}$, then you pick a root of some (monic) polynomial in $\mathbb{Q}[x]$. You could equally well take that polynomial to have integral coefficients by clearing denominators (as this doesn't change the roots), and then take its reduction modulo a prime $p$, and adjoin a root of the resulting element of $\mathbb{F}_p[x]$ to get an extension of $\mathbb{F}_p$.

One issue with this is that, when you cleared denominators, the resulting integral coefficients could have been partially/entirely divisible by $p$, so the resulting extension of finite fields could bear little resemblance to the extension of the rationals. One can get around this by restricting to algebraic integers, as opposed to algebraic numbers. This simply means that you start with a monic polynomial over $\mathbb{Z}$, and then any root of that can be used to make an extension of $\mathbb{F}_p$, or any field for that matter. This perspective is fundamental in algebraic number theory.

If you want to adjoin a single transcendental number to $\mathbb{Q}$, then as a field extension that's just isomorphic to $\mathbb{Q}(t)$, so perhaps the most reasonable analog over the finite field would just be $\mathbb{F}_q(t)$. Basically, the point is that from the perspective of solely field theory, $\pi$ is not much different than any other transcendental.

A rather different way one might address your question is by discussing the field called $\mathbb{C}_p$, which is the completion of the algebraic closure of the $p$-adic field $\mathbb{Q}_p$. Coupled with what are called Teichmuller representatives, and an application of the axiom of choice, one actually can "make sense" of something like $zm$ for $z\in \mathbb{C}$ and $m\in \mathbb{F}_p$, but I should acknowledge that that is not a useful way to think about things (as far as I know).