Is there an 'unnatural' topological construction of an algebraically closed field of positive characteristic?

Question

It's well known that while there is a natural topological construction of a nearly algebraically closed field of characteristic $0$, algebraically closed fields of positive characteristic seemingly need to be constructed by a purely algebraic process (which is of course natural in its own way, but is of a very different character from the construction of $\mathbb{R}$ from $\mathbb{Q}$). Even among non-Archimedean local fields, there's a stark contrast between the characteristic $0$ case and the positive characteristic case: $\mathbb{Q}_p$ contains many irrational algebraic numbers and moreover the absolute Galois group of $\mathbb{Q}_p$ is topologically finitely generated (like how the absolute Galois group of $\mathbb{R}$ is finite) but $\mathbb{F}_p((t))$ is essentially no closer to being algebraically closed than $\mathbb{F}_p$ is. Relatedly, in the natural topology on $\overline{\mathbb{F}_p((t))}$ induced by the norm, $\overline{\mathbb{F}_p}$ is uniformly discrete.

Because of this I was surprised to learn recently that one can construct a (Hausdorff) field topology on $\overline{\mathbb{F}_p}$ with the property that every infinite subfield is non-discrete. (Unfortunately I can't find the reference at the moment. It was a thesis rather than a paper in a journal.) This is of course a careful by-hand construction rather than something more natural like order completion or completion with regards to an absolute value, but this lead me to wonder about the extent to which there really is a fundamental obstruction to adding roots of polynomials to $\mathbb{F}_p$ by some kind of topological completion.

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This is a fairly open-ended question, but I also have some more specific versions of it. I think that part of what makes the construction of $\mathbb{R}$ from $\mathbb{Q}$ feel so different is that it's easy to motivate the construction in a way that doesn't seem to be about building new algebraic numbers, yet many algebraic numbers show up. It probably doesn't make sense to want to do something like this with a finite field, but the field of fractions $\mathbb{F}_p(t)$ seems like a reasonable substitute for $\mathbb{Q}$.

Question 1. Is there a field $F$ that is in some sense a 'topological completion' of $\mathbb{F}_p(t)$ in which there are elements of $F$ non-trivially algebraic over $\mathbb{F}_p$?

Arguably the friendliest feeling topological spaces are metric spaces, which could motivate a more specific version of this question. Recall that a topological ring $(R,+,\cdot)$ is a topological space $R$ together with continuous ring operations $+ : R^2 \to R$ and $\cdot : R^2 \to R$, and a topological field is a topological ring $(F,+,\cdot)$ that is a field which moreover satisfies that $x \mapsto x^{-1}$ is continuous on $F \setminus \{0\}$. Recall that a Polish space is a completely metrizable space.

Question 2. Is there a topological ring $(F,+,\cdot)$ such that $F$ is a Polish space, $F$ admits a dense field embedding from $\mathbb{F}_p(t)$ (or $\mathbb{F}_p(t_1,\dots,t_n)$ for some finite $n$), and there are elements of $F$ that are non-trivially algebraic over $\mathbb{F}_p$? Can such an $F$ be a topological field? Can such an $F$ have an absolute Galois group that is topologically finitely generated or even finite? Can such an $F$ actually be algebraically closed?

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I think another reason that $\mathbb{R}$ and $\mathbb{C}$ (and even to some extent the $p$-adics once you get used to them) feel so much more concrete is that they can be pictured in a relatively clear way. Obviously since $\overline{\mathbb{F}_p}$ is countable, there's a clear topological picture of what it looks like but this picture has no algebraic information. It's well know that there's a picture of the multiplicative group $\overline{\mathbb{F}_p}^\times$ in that it can be (non-canonically) embedded into the circle group $S^1$ (specifically hitting the $\ell$th roots of unity for $\ell$ coprime to $p$), so it might be a little bit reasonable to picture $\overline{\mathbb{F}_p}$ as a circle plus an isolated point for $0$. The issue of course is that this picture contains no real information about addition. The intuition I've gotten from some conversations with number theorists (and some results in the model theory of fields with multiplicative circular orders) is that in some sense addition on $\overline{\mathbb{F}_p}$ is 'random' relative to this picture. Regardless it seems reasonable to try to massage this into some kind of 'visualizable' construction of $\overline{\mathbb{F}_p}$.

Question 3. Fix a prime $p$. Let $G \subseteq S^1$ be the group of $\ell$th roots of unity for $\ell$ coprime to $p$. Is there an easy-to-describe (necessarily discontinuous) operation on $+: (S^1 \sqcup \{0\})^2 \to S^1 \sqcup \{0\}$ such that the restriction to $G \sqcup \{0\}$ is addition on $\overline{\mathbb{F}_p}$? If not, is there some way of 'randomly' or 'generically' selecting such an operation $+$ such that it almost surely restricts to addition on $\overline{\mathbb{F}_p}$? If either of these are possible, how algebraically nice can $+$ be on $S^1 \sqcup \{0\}$? Can it form a (semi-)ring?

I think it's also reasonable to ask how topologically nice such an operation could be but it's easy to show that with no other restrictions it can be Borel (since $G$ is countable), but in the presence of other algebraic restrictions this seems to be non-trivial.

Question 4. If the operation $+$ in Question 3 makes $S^1 \sqcup \{0\}$ into a ring or semi-ring, can it be Borel?

Answer 1

This is not directly an answer to any of your questions as stated but a riff on the theme of "what does $\overline{\mathbb{F}_p}$ look like?" The best answer to this question I've found so far comes from a generalization of the normal basis theorem to infinite Galois extensions due to Lenstra, which when applied to $\overline{\mathbb{F}_p}$ asserts that $\overline{\mathbb{F}_p}$, as a module over its Galois group $\widehat{\mathbb{Z}}$, can be identified with the regular module of continuous functions $\widehat{\mathbb{Z}} \to \mathbb{F}_p$.

Unwrapping this description to make it much more concrete: suppose we temporarily give up on understanding the full multiplication on $\overline{\mathbb{F}_p}$ and settle for understanding its addition, together with the action of the Frobenius map $F : x \mapsto x^p$. The normal basis theorem applied to $\mathbb{F}_{p^n}$ answers this question for $\mathbb{F}_{p^n}$: it tells us that we can find $\alpha_n \in \mathbb{F}_{p^n}$ such that $\alpha_n, F(\alpha_n), \dots F^{n-1}(\alpha_n)$ is a basis of $\mathbb{F}_{p^n}$. If we write elements of $\mathbb{F}_{p^n}$ in terms of this basis, this means we can understand the action of the Frobenius map on $\mathbb{F}_{p^n}$ as being isomorphic to the action of the cyclic shift operator acting on strings of length $n$ over the alphabet $\mathbb{F}_p$. This lets us identify Galois orbits on $\mathbb{F}_{p^n}$ with necklaces, which came up in this old MO question of mine.

Among other things this offers an elegant view of how the finite fields nest into each other: for every $d \mid n$ the subfield $\mathbb{F}_{p^d}$ of $\mathbb{F}_{p^n}$ gets identified with words of length $n$ which are periodic with period dividing $d$ (since these are exactly the fixed points of $F^d$).

Consider, for example, $p = 2, n = 4$: here we can think of the $16$ elements of $\mathbb{F}_{16}$ as the $16$ binary strings $0000, 0001, \dots$ of length $4$, with the $4$ elements of $\mathbb{F}_4$ nested inside as the $4$ binary strings $0000, 0101, \dots$ of length $4$ and period $2$, and with the $2$ elements of $\mathbb{F}_2$ nested inside these as the $2$ binary strings $0000, 1111$ of length $4$ and period $1$. This means that $\alpha_4 = 1000$, whatever it is, must have the property that $\alpha_4 + F^2(\alpha_4) = 1010 = \alpha_2 \in \mathbb{F}_4$, where $\alpha_2 + F(\alpha_2) = 1111 \in \mathbb{F}_2$ is the multiplicative identity. So we may not know everything about the multiplication but we do know some things.

At this point it ought to be tempting to wonder: can we make this identification consistently across $\mathbb{F}_{p^n}$ for all $n$ simultaneously? Lenstra's normal basis theorem implies that we can: it allows us to identify $\overline{\mathbb{F}_p}$ itself with the vector space of infinite periodic strings over the alphabet $\mathbb{F}_p$, where the Frobenius acts by a shift, and hence identifies $\mathbb{F}_{p^n}$ with the subset of strings of period dividing $n$. So a string of period exactly $n$ corresponds to an algebraic number over $\mathbb{F}_p$ of degree $n$. I find this picture very elegant - IMO it makes the nesting structure of the finite fields very vivid and concrete - although again we've sacrificed a full understanding of the multiplication to get it.

This suggests a variation of your questions 3 and 4, which is whether there is a reasonable description of the full multiplication in terms of these periodic strings. I would be surprised if there were, but I think either a positive or negative answer would be pretty interesting.

Another variant question: forget trying to construct all of $\overline{\mathbb{F}_p}$ at once, can we even construct $\mathbb{F}_{p^n}$ for any $n \ge 2$ without having to choose a specific irreducible polynomial of degree $n$ (or make some other much worse choice, such as a choice of algebraic closure of $\mathbb{F}_p$)? The closest I've gotten to this so far is that $\mathbb{F}_p[x]/\Phi_{p^n-1}(x)$ is a product of many copies of $\mathbb{F}_{p^n}$, one for each Galois orbit of primitive element.