Timeline for Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?
Current License: CC BY-SA 2.5
11 events
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| Apr 19, 2021 at 19:13 | comment | added | Tim Campion | One obstruction to having a functorial construction of algebraic closures is the fact that the classifying space of the category of field extensions of $k$ is not typically homotopy equivalent to the category of algebraically closed field extensions of $k$ -- the former is contractible while the latter is $BGal(\bar k / k)$. To the extent that constructivism goes hand in hand with functoriality, and to the extent that this obstruction is encoded in the Galois group, I wonder if the answer to this question might be dependent on the inverse Galois problem... | |
| Nov 20, 2010 at 12:01 | answer | added | Joel David Hamkins | timeline score: 32 | |
| Nov 19, 2010 at 22:39 | answer | added | Eivind Dahl | timeline score: 4 | |
| Nov 19, 2010 at 9:27 | vote | accept | Qiaochu Yuan | ||
| Nov 19, 2010 at 2:18 | comment | added | Andrés E. Caicedo | Hi Timothy. Thanks for finding these posts, I thought they were more recent and couldn't find them. | |
| Nov 19, 2010 at 2:10 | comment | added | Timothy Chow | Not exactly what you asked, but there was some discussion on the Foundations of Mathematics mailing list about whether the existence of an algebraic closure of Q is equivalent to the assumption that a countable union of finite sets is countable. See this post by Harvey Friedman cs.nyu.edu/pipermail/fom/2006-May/010541.html and this post by Andreas Blass cs.nyu.edu/pipermail/fom/2006-May/010551.html pointing out some difficulties. | |
| Nov 19, 2010 at 1:31 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Nov 19, 2010 at 1:29 | comment | added | Willie Wong | (BTW: credit where credit's due: I learned of that website from Andres mathoverflow.net/questions/45928/… ) | |
| Nov 19, 2010 at 1:29 | answer | added | Andrés E. Caicedo | timeline score: 25 | |
| Nov 19, 2010 at 1:28 | comment | added | Willie Wong | Every field has an algebraic closure is "Form 69" in consequences.emich.edu/conseq.htm , and it doesn't list any equivalent forms to it. | |
| Nov 19, 2010 at 1:10 | history | asked | Qiaochu Yuan | CC BY-SA 2.5 |