Timeline for What are the necessary conditions for a real number to be a cyclotomic integers？
Current License: CC BY-SA 3.0
21 events
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| Oct 30, 2015 at 17:53 | vote | accept | Xiao-Gang Wen | ||
| Oct 28, 2015 at 13:56 | comment | added | Xiao-Gang Wen | Indeed, I need to learn what is a "totally real cyclotomic integer". Thanks, Scott! When I ask a question on MO, what I usually gained is how to ask the right question :-) | |
| Oct 28, 2015 at 4:35 | comment | added | Kim Morrison | Addressing all of the comments above: Xiao-Gang didn't realize he meant to ask about totally real cyclotomic integers. With this, it becomes quite interesting. | |
| Oct 28, 2015 at 4:33 | answer | added | Kim Morrison | timeline score: 7 | |
| Oct 27, 2015 at 22:44 | answer | added | Dave Penneys | timeline score: 10 | |
| Oct 27, 2015 at 16:39 | comment | added | Xiao-Gang Wen | @Emil: Thanks for the comment. Since real cyclotomic integers are dense, we are facing a similar issue as in testing if a real numer is a rational number or not. So we need to introduce some similicity conditions. But this requires me to refer to fusion category. I am considering fusion category (FC) of finite rank $N$. If $N$ is not too large, what real numbers can be the quantum dimensions of the FC? quantum dimensions are cyclotomic integers (may be "simple" ones in some sense), | |
| Oct 27, 2015 at 16:31 | comment | added | P Vanchinathan | @Emil: Expanding on your comment integers of all quadratic fields, following Kronecker-Weber (or something simpler like quadratic reciprocity), are cyclotomic integers. | |
| Oct 27, 2015 at 14:26 | comment | added | Emil Jeřábek | Well, $\mathbb Z+\sqrt2\mathbb Z$ is a set of cyclotomic integers dense in $\mathbb R$, hence you can approximate arbitrary reals by those. | |
| Oct 27, 2015 at 12:47 | history | edited | Xiao-Gang Wen | CC BY-SA 3.0 |
added 235 characters in body
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| Oct 27, 2015 at 11:43 | comment | added | Xiao-Gang Wen | I made some change. Now the question is about Cyclotomic integers. | |
| Oct 27, 2015 at 11:40 | history | edited | Xiao-Gang Wen | CC BY-SA 3.0 |
deleted 7 characters in body; edited title
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| Oct 27, 2015 at 10:36 | comment | added | GH from MO | I think you should delete your first sentence, and make a remark that a necessary condition (for your second sentence) is that the number is an algebraic integer. Indeed, testing whether a positive real number is an algebraic integer is much like testing whether it is an integer. In the end you have the definition, not a test. | |
| Oct 27, 2015 at 3:31 | history | edited | Xiao-Gang Wen | CC BY-SA 3.0 |
added 4 characters in body
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| Oct 27, 2015 at 2:27 | comment | added | P Vanchinathan | A real number $x$ is an algebraic integer if and only if $-x$ is. So positivity plays no role. | |
| Oct 26, 2015 at 23:53 | comment | added | Xiao-Gang Wen | @David Speyer: Thank you very much. Your reference is very helpful. | |
| Oct 26, 2015 at 23:24 | comment | added | David E Speyer | It sounds like you might be interested in arxiv.org/abs/1004.0665 , as well as many of the other recent papers of Scott Morrison and Noah Snyder front.math.ucdavis.edu/… | |
| Oct 26, 2015 at 23:10 | comment | added | Fan Zheng | Have you tried stack exchange? | |
| Oct 26, 2015 at 23:04 | history | edited | Xiao-Gang Wen | CC BY-SA 3.0 |
edited title
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| Oct 26, 2015 at 21:34 | comment | added | Xiao-Gang Wen | "What sort of answer are you expecting?" Like the example given in the second paragraph of the question. ie something that can be checked easily. | |
| Oct 26, 2015 at 19:45 | comment | added | Geoff Robinson | What sort of answer are you expecting? A positive real number is an algebraic integer if and only if it is a root of a monic polynomial with integer coefficients. | |
| Oct 26, 2015 at 19:10 | history | asked | Xiao-Gang Wen | CC BY-SA 3.0 |