User cristos a. ruiz - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:01:12Z http://mathoverflow.net/feeds/user/4619 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/129804/galois-group-of-constructible-numbers Galois group of constructible numbers Cristos A. Ruiz 2013-05-06T06:46:20Z 2013-05-07T02:36:55Z <p>Let $\mathcal{C}$ be the field of constructible numbers, that is, the complex numbers constructible by compass and straightedge. It can be shown that it consists of all the numbers obtainable by adding square roots iteratively to the rational numbers, and it's easy to see that the extension $\mathcal{C}/\mathbb{Q}$ is Galois.</p> <p>Can anything be said about the group $\mathrm{Gal}(\mathcal{C}/\mathbb{Q})$?</p> <p>For example, it has a quotient isomorphic to the multiplicative group of invertible $2$-adic numbers since it contains the $2^n$ roots of unity.</p> <p><strong>Edit:</strong> The biggest quotient I can think of explicitly is the one given by the subextension $\mathbb{Q}(\sqrt[2^\infty]{\mathbb{Q^*}})$ (add all the $2$-power roots of every non-zero rational) which by Kummer theory has Galois group isomorphic to $\mathbb{Z}_2^\infty\rtimes\mathbb{Z}_2^*$.</p> <p>The question is analogous if one changes the prime $2$ by any prime $p$. I just liked the relationship with compass and straightedge constructions in the case $p=2$.</p> http://mathoverflow.net/questions/28967/a-characterization-of-convexity A characterization of convexity Cristos A. Ruiz 2010-06-21T16:23:24Z 2013-05-06T07:15:14Z <p>While doing some research on polytopes I came to the following question. Maybe it's already somewhere but anyway I'll post it here.</p> <p>Let $X\subset \mathbb{R}^3$ be such that, for every plane $P$, $P\cap X$ is simply connected. Is $X$ convex?</p> <p>I'm not sure, but I think maybe it's necesary to assume some well behaving like local simple connectedness. Anyway I think this is true with the apropiate asumptions. I would not be surprised if it was true just as stated.</p> <p>Probably this is true even in greater dimensions.</p> http://mathoverflow.net/questions/19046/open-source-mathematical-software Open source mathematical software. Cristos A. Ruiz 2010-03-22T18:46:15Z 2012-09-21T09:07:53Z <p>I want some recomendation on which software I should install on my computer, an open source program for general abstract mathematical purposes (as opposed to applied mathematics). </p> <p>I would likely use it for group theory, number theory, algebraic geometry and probably polytopes. </p> <p>The kind of program I have in mind is Mathematica or Matlab. Altough probably those are not designed por abstract mathematics.</p> <p>Any suggestions?</p> http://mathoverflow.net/questions/18833/jordan-holder-decomposition-for-group-objects Jordan Hölder decomposition for group objects Cristos A. Ruiz 2010-03-20T12:51:13Z 2010-04-11T19:41:23Z <p>Is there some generalization of the Jordan-Hölder decomposition for group objects in a category $\mathcal{C}$?</p> <p>If $\mathcal{C}$ is the category Sch$(S)$ of schemes over a base scheme $S$ then (I think) this is true, also probably for other categories of "spaces" like Top or Diff it should be true, but I don't have any idea for general categories.</p> http://mathoverflow.net/questions/18666/continuous-function-from-0-1-to-0-1 Continuous function from $[0,1]$ to $[0,1]$ Cristos A. Ruiz 2010-03-18T21:59:37Z 2010-03-18T22:04:35Z <p>Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?</p> http://mathoverflow.net/questions/18433/is-there-a-non-trivial-topological-group-structure-of-mathbbz Is there a non-trivial topological group structure of $\mathbb{Z}$? Cristos A. Ruiz 2010-03-16T22:19:53Z 2010-03-16T22:39:00Z <p>More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?</p> http://mathoverflow.net/questions/130511/diameter-area-ratio-for-affine-tranformations Comment by Cristos A. Ruiz Cristos A. Ruiz 2013-05-13T21:02:36Z 2013-05-13T21:02:36Z I guess $F$ must have positive area. This is trivialy false if $F$ is a line segment. http://mathoverflow.net/questions/129804/galois-group-of-constructible-numbers/129828#129828 Comment by Cristos A. Ruiz Cristos A. Ruiz 2013-05-06T20:56:28Z 2013-05-06T20:56:28Z Thanks for pointing me to this bibliography, I have a lot to read now. http://mathoverflow.net/questions/129804/galois-group-of-constructible-numbers Comment by Cristos A. Ruiz Cristos A. Ruiz 2013-05-06T20:55:24Z 2013-05-06T20:55:24Z Thanks @Chandan Singh Dalawat and @Fran&#231;ois Brunault for your enlightening comments. http://mathoverflow.net/questions/129804/galois-group-of-constructible-numbers Comment by Cristos A. Ruiz Cristos A. Ruiz 2013-05-06T08:55:29Z 2013-05-06T08:55:29Z @Brunault so (correct me if I'm wrong) it must contain the field generated by the Tate module $T_2(A)$ for every abelian variety $A$ of dimension $2n$ such that the Galois group is $\mathrm{GL}_{2n}(\mathbb{Z}_2)$. That looks like a nice result. http://mathoverflow.net/questions/129804/galois-group-of-constructible-numbers Comment by Cristos A. Ruiz Cristos A. Ruiz 2013-05-06T08:46:26Z 2013-05-06T08:46:26Z @Chandan Singh Dalawat Yes, in fact $\mathrm{Gal}(K_{n+1}/K_n)$ is isomorphic to a countable product of copies of the group of order two. But already $\mathrm{Gal}(K_2/K_0)$ seems difficult to me, it properly contains the field $K_0(\sqrt[4]{K_0})$ which has Galois group over $K_0$ isomorphic to the semidirect product of $\mathbb{Z}/2\mathbb{Z}$ by a countable product of copies of $\mathbb{Z}/4\mathbb{Z}$ http://mathoverflow.net/questions/129804/galois-group-of-constructible-numbers Comment by Cristos A. Ruiz Cristos A. Ruiz 2013-05-06T07:02:08Z 2013-05-06T07:02:08Z I should have written complex numbers instead of lengths since length does mean positive real numbers, I just edited accordingly. I meant to include all those numbers, the last paragraph wouldn't be right with my original definition of $\mathcal{C}$. http://mathoverflow.net/questions/28967/a-characterization-of-convexity/28987#28987 Comment by Cristos A. Ruiz Cristos A. Ruiz 2010-06-24T18:28:46Z 2010-06-24T18:28:46Z I'm chosing this one as the correct answer. There was also Matessi's correct counterexample and Pak's reference (by the way, that book looks really good). But I think there's more merit in this one, and it looks correct to me, also I don't know if the case of $X$ open was known, and the method used to solve it is very appealing to me. Thanks very much for your responses! http://mathoverflow.net/questions/28967/a-characterization-of-convexity Comment by Cristos A. Ruiz Cristos A. Ruiz 2010-06-21T22:41:08Z 2010-06-21T22:41:08Z n-1 planes and yes, connected and simply connected http://mathoverflow.net/questions/18833/jordan-holder-decomposition-for-group-objects Comment by Cristos A. Ruiz Cristos A. Ruiz 2010-03-22T18:34:43Z 2010-03-22T18:34:43Z I mean the following: if $S$ is a subgroup and $N$ a normal subgroup of $G$ then $N$ is normal in $NS$, $N\cap S$ is normal in $S$ and $NS/N$ is isomorphic to $S/N\cap S$. http://mathoverflow.net/questions/18833/jordan-holder-decomposition-for-group-objects Comment by Cristos A. Ruiz Cristos A. Ruiz 2010-03-20T17:02:53Z 2010-03-20T17:02:53Z That's exactly what I meant, given that there exists a composition series, it should be unique up to permutation. After thinkingof it for a while, I believe that it should be true for any category in which the second isomorphism theorem holds. http://mathoverflow.net/questions/18666/continuous-function-from-0-1-to-0-1/18668#18668 Comment by Cristos A. Ruiz Cristos A. Ruiz 2010-03-18T22:14:13Z 2010-03-18T22:14:13Z Great I didn't thought of that one, can I ask differentiability? http://mathoverflow.net/questions/18588/is-a-inverse-limit-of-compact-spaces-again-compact Comment by Cristos A. Ruiz Cristos A. Ruiz 2010-03-18T13:57:22Z 2010-03-18T13:57:22Z Perhaps you want to say compact in place of closed where you say &quot;this product is closed by Tychonoff&quot; http://mathoverflow.net/questions/18433/is-there-a-non-trivial-topological-group-structure-of-mathbbz/18435#18435 Comment by Cristos A. Ruiz Cristos A. Ruiz 2010-03-16T22:48:51Z 2010-03-16T22:48:51Z Sorry, I had those examples and related ones in mind and wanted to prove they were the only ones. I'll think of a well-posed question and post it again.