Question
What's $$\lim_{n\to\infty} GF(2^n)$$ and does it equal $\Bbb Q_2$?
Or if that doesn't make sense, some correctly conceived limit which fits the obvious motivation.
I'm expecting say the field of 2-adic numbers $\Bbb Q_2$.
My Attempt - and a little more clarity
One one hand I think I need the limit to be well-defined in order for the question to have a unique answer. On the other I'm expecting the only reasonable answer to be $\Bbb Q_2$, so I need to ask in what metric or limit I would get $\Bbb Q_2$. The 2-adic metric seems the obvious choice in which to define the limit but I don't know if that gives me $\Bbb Q_2$.
A really nice answer here describes building an ultrafilter and then quotient out by some maximal ideal but there's quite a lot to parse there for me to make it practical for me to execute. Aside from the fact it seems to say there are many choices of how you do it, I can't see (with my limited capabilities) any reason why one of those must or must not correspond with $\Bbb Q_2$.
Motivation
I have a particular motivation for the original question which may inform the choice or definition of limit. OEIS A001037 enumerates the degree-$n$ irreducible polynomials over $GF(2)$. It turns out these polynomials are in bijection with the cyclic points of period $n$ of the Collatz conjecture over $\Bbb Q_2$. So I'd like to close the circle by creating the field which contains all those irreducible polynomials as well as all of the corresponding cyclic points (and is therefore also $\Bbb Q_2$) .
Maybe it's a corrollary of this motivation, that I need a limit or topology which somehow respects the degree $n$ irreducible polynomials over $GF(2)$? I'm not sure how that would work, and what "respects" would mean. These are in bijection with the base $2$ Lyndon words of length $n$ so I guess the algebras on Lyndon words could come into play.