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I added a new section to algebraically closed field, on the classification in terms of characteristic and transcendence degree.
algebraically closed field so far does not say anything on the usual facile question of
I think some short section on this should be added. Moreover, there appear to be something to be said here connected to unnatural isomorphism.
I could say more about this, but: it is likely that some readers round here know more field theory than I do (and in particular could make the section somewhat “constructive-field-theory-informed”), so it seems more efficient communication to first ask whether someone else fields like editing algebraically closed field accordingly.
It is a theorem that there is no such endofunctor exists that renders the inclusion $F \to \widebar{F}$ natural. Yes, this is at least worth a remark at algebraically closed field.
In fact something was said about this already at unnatural isomorphism, both at “Between objects” and in 5. at Examples.
I have now added an Example at algebraically closed field that illustrates what goes wrong.
Thanks for the response.
SVGsandbox contains what I think is a relevant diagram (more or less self-explanatory; AC is an intentional pun on “axiom of choice”).
Reason: I had to test something TikZ-technical, and thought I could just as well test it on something meaningful.
Thinking this lab-like style, I put it in that sandbox.
If someone else can make instructive use of the diagram, either in algebraically closed field or in unnatural isomorphism, I’d be delighted to see it.
If not, I will remove it from the sandbox in due course.
This topic is not what I have to mainly work on currently, but—if it does not get out of hand— I will gladly modify the diagram on request, if this is necessary to make it more usable.
Sorry, but to be honest, I find it hideous, practically unreadable even, in its current form. For example, does the word “Field” in that enormous font have to be there?? Same with the enormous arrows.
Thanks for the response. Like I said, I was testing something.
Note: I accidentally wrote something highly ambiguous: with
Thinking this lab-like style, I put it in that sandbox.
I did in no way mean to say that the illustration was in nLab style, but rather I meant this particular way of working, i.e., that maybe, now and then, in certain situations, dropping by-products of one’s work, while simultaneously asking in the forum whether someone has a relevant use for it for the nLab, seems like some “labs” function.
In the unlikely case that you would like to use it somewhere (for example if you think there is something canonical to say about possible “canonical definitions” of 2-cells between the composites…), then please specify what should be changed. Not that it would be important to me to use that illustration somewhere.
I can’t foresee any use I would have for that diagram, but generally speaking curvy arrows would be useful if they are easily available. But this topic is outside this thread (and would belong under the Technical Matters category).
Relatedly, the algebraic closure of a field is not definable in the theory of fields (in the sense that there is no definable functor—one isomorphic to an evaluation functor by an object of $\mathbf{Def}(T_{\operatorname{fields}})$—from $\mathbf{Mod}(T_{\operatorname{fields}}) \to \mathbf{Set}$ taking fields to their algebraic closures.) This is because such a functor would have to commute with ultraproducts, but taking algebraic closures doesn’t. For example, $\left(\mathbb{Q}^{\mathcal{U}}\right)^{\operatorname{alg}}$ is contained in $\left(\mathbb{Q}^{\operatorname{alg}}\right)^{\mathcal{U}}$ (the inclusion $\mathbb{Q} \hookrightarrow \mathbb{Q}^{\operatorname{alg}}$ is not elementary, but at the level of sets induces an injective comparison map between colimits anyways), but if $\{\zeta_n\}_{n \in \mathbb{N}}$ are primitive $n$th roots of unity, then their ultralimit $[\zeta_n]_{n \to \mathcal{U}}$ in $\left(\mathbb{Q}^{\operatorname{alg}}\right)^{\mathcal{U}}$ is transcendental over $\mathbb{Q}^{\mathcal{U}}$.
Moved the discussion on the Foundations of Mathematics mailing list and the Banaschewski and Richman papers to a newly created references section and added the Ruitenberg paper which also gives a definition of a field being algebraically closed. In addition, the Banaschewski and Richman papers are now properly referenced:
{#Banaschewski92} Bernhard Banaschewski, Algebraic closure without choice, Mathematical Logic Quarterly, Volume 38, Issue 1, 1992, Pages 383-385, [doi:10.1002/malq.19920380136]
{#Richman00} Fred Richman, The fundamental theorem of algebra: a constructive development without choice, Pacific Journal of Mathematics 196 1 (2000) 213–230 [doi:10.2140/pjm.2000.196.213, pdf]
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