Timeline for uncountable algebraically closed field other than C ?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Jul 7, 2015 at 14:07 | comment | added | Lazzaro Campeotti | Dear Pete, thanks a lot for the reference and those keywords. (In your notes, the relevant result Corollary 111 has a broken reference to an earlier result (Proposition ??).) | |
| Jul 7, 2015 at 14:05 | comment | added | Pete L. Clark | @potentially dense: See $\S$12.2 of math.uga.edu/~pete/FieldTheory.pdf. Alternately, as the comments describe, from a model theoretic perspective this is a consequence of the model completeness of the theory of algebraically closed fields, itself a corollary of quantifier elimination in such fields. For that perspective, see $\S$5.4 of math.uga.edu/~pete/modeltheory2010FULL.pdf. (Or better, see any introductory model theory text whatsoever...) | |
| Jul 7, 2015 at 10:17 | comment | added | Lazzaro Campeotti | Dear Pete, do you know a convenient reference for the statement in the first sentence of your second paragraph? A bit of Googling reveals it is called Steinitz' (third) theorem; I found Steinitz' original paper, but I would like something a bit more digestible to refer to. | |
| Aug 24, 2013 at 21:33 | comment | added | Metin Y. | Just to add the terminology to it: The theory of algebraically closed fields of characteristic $p$, $ACF_p$, is $\kappa$-categorical for uncountable $\kappa$'s. | |
| Nov 29, 2011 at 22:03 | comment | added | Theo Johnson-Freyd | @Pete: Thanks for the comments --- they are very helpful. Indeed, being neither a set theorist nor an algebraic geometer, I wasn't aware of the "roomy enough" results (although I am not surprised by them). | |
| Nov 29, 2011 at 13:28 | comment | added | Pete L. Clark | It seems that the only "familiar" operation which takes us beyond the continuum is formation of the power set...but taking unrestricted power sets is not something you need to do in field theory. Well, maybe I've just pushed the issue around the landscape a bit. I hope you found these comments at least somewhat helpful. | |
| Nov 29, 2011 at 13:26 | comment | added | Pete L. Clark | But that won't even get you to continuum cardinality. For that I think we have to talk about topological completion: you start with a countably infinite topological space and realize that in some sense you are missing some points, so you "complete" it (in any of various ways) and tend to get a space of continuum cardinality. For instance the completion of a countably infinite separable metric space without isolated points will necessarily be of continuum cardinality. | |
| Nov 29, 2011 at 13:23 | comment | added | Pete L. Clark | @Theo: (Well, of course I'm not a set theorist either.) There are various ways to try to justify this. For instance, in Weil-style algebraic geometry it was recognized that an algebraically closed field of (not even necessarily uncountably) infinite absolute transcendence degree is roomy enough to encompass all needed geometric constructions. The language of the time was universal domain, but I think a better take on it is that such fields are precisely the saturated models of the theory of algebraically closed fields, so are in a precise sense sufficiently rich. | |
| Nov 29, 2011 at 7:54 | comment | added | Theo Johnson-Freyd | I agree that I have never had to think about infinite sets of cardinalities other than countable and continuum. (Occasionally, I hear about crank mathematicians who want to claim that we should work in a version of "set theory" in which this is all there are.) But, not being a set theories, I have no understanding of why this might be. Do you have any stories you tell yourself for why there are no specific reasons to algebraically closed fields of larger than continuum cardinality? Or do you also only have the observation that there seems to be no interesting math in those examples? | |
| May 20, 2010 at 11:51 | vote | accept | Laurent | ||
| May 20, 2010 at 11:30 | vote | accept | Laurent | ||
| May 20, 2010 at 11:51 | |||||
| May 20, 2010 at 11:24 | history | answered | Pete L. Clark | CC BY-SA 2.5 |