Fundamental Theorem of Algebra for fields other than $\Bbb{C}$, or how much does the Fundamental Theorem of Algebra depend on topology and analysis?

When proving the Fundamental Theorem of Algebra, we need to appeal to analytic and/or topological properties of $\Bbb{C}$ and $\Bbb{C}[z]$. Is this going to be necessary in general, and if so, to what extent?

That is, suppose a field $K$ is given, and we desire to show that $K$ is algebraically closed. Is there any amount of purely algebraic data (save that $K = F^{alg}$ for some field $F$) that will allow us to say that $K$ is algebraically closed? Of course, the phrase "purely algebraic data" isn't well-defined, but I loosely mean information about $K$ in relation to fields $E_i$ of which $K$ is an extension, such as the degrees of the extensions $K/E_i$, if $K = E_i(\alpha)$ for some $\alpha\in K$, their Galois groups $\operatorname{Gal}\left(K/E_i\right)$, what $\operatorname{char}K$ is, and so on (where this information isn't precisely the information that $K$ is constructed as the algebraic closure of some field).

If this isn't possible, at what point does it become necessary to appeal to the topology and analysis of $K$ and $K[x]$, and how does this "point" depend on $K$? I realize that exactly how and where analysis and topology come into play for different $K$ will depend on the nature of the $K$ with which one is working, but I would be interested in knowing what properties of $K$ have the greatest effect on the necessity of analysis and topology in a proof that $K$ is algebraically closed. Of course, the more general these properties, the better.

A note: this question deals with this idea to some extent, although the discussion there is more focused on $\Bbb{C}$, and I would like to consider a more general setting: the conditions required to show a certain type of field is algebraically closed, and how those conditions differ for different types of fields.

this is borrowed from the comments in the link to the question you posted:

Let $F$ be a field of characteristic $0$, and $K / F$ be a finite Galois extension. Suppose every polynomial of odd degree in $F[x]$ has a root in $F$, and every polynomial of degree $2$ in $K[x]$ has a root in $K$. Then $K$ is algebraically closed.

Also, it turns out that algebraically closures are rarely finite extensions. In fact, if $F$ is a field and $C / F$ a finite extension that is algebraically closed, then $C=F(i)$ where $i^{2}=-1$, and $F$ has characteristic $0$. Also for every nonzero $a \in F$, either $a$ or $-a$ is a square, and finite sums of nonzero squares are nonzero squares. The source for this is Keith Conrad's notes:

http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/artinschreier.pdf

There’s one other way of getting an algebraically closed field: start with $\mathbb F_p=\mathbb Z/p\mathbb Z$, and adjoin an $m$-th root of unity for every natural number $m$ relatively prime to $p$. In other words, adjoin all possible roots of unity to $\mathbb F_p$. That’s the construction, and now, of course, there’s the exercise of showing that the result is indeed algebraically closed.