Timeline for Why do they think this question is not of research level?
Current License: CC BY-SA 3.0
24 events
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| Aug 23, 2013 at 20:15 | comment | added | Makoto Kato |
@ToddTrimble [@MakotoKato Well, if you already understand all this, I now join the rest in wondering what the problem is. Did Michael Zieve answer your query satisfactorily at MSE?] I edited my question to make it clearer. I'm not sure Michael Zieve's answer at MSE is finite in nature. Please don't think that I edited it to evade his answer. The edited version is what I had originally in my mind.
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| Aug 23, 2013 at 19:17 | comment | added | Todd Trimble | @MakotoKato Well, if you already understand all this, I now join the rest in wondering what the problem is. Did Michael Zieve answer your query satisfactorily at MSE? | |
| Aug 23, 2013 at 19:14 | comment | added | Todd Trimble | @YemonChoi Agreed, definitely not research level, and definitely a better fit for MSE than MO (all the more so if less than perfectly posed). | |
| Aug 23, 2013 at 19:13 | comment | added | Makoto Kato | @ToddTrimble I know the basics of ordered fields written , for example, in Bourbaki. | |
| Aug 23, 2013 at 19:07 | comment | added | Makoto Kato |
@ToddTrimble [This fact is independent of whether or not the OP knows the formal details (and I think it would be more productive not to badger the OP over these, since he/she might not know them, but to help him/her by incorporating them into an answer).] Just in case you are wondering, I'm male. I opened a meta thread in MSE about this problem.
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| Aug 23, 2013 at 19:07 | comment | added | Todd Trimble | Noah: how "obvious" all this was to him is hard to say, but it seems he wanted to get at a proof of the positivity without availing himself of the real number field, and this is where Michael's answer (outlining a proof of orderability of $L$) presumably fulfills something that had been missing for him. | |
| Aug 23, 2013 at 17:17 | comment | added | Noah Snyder | @ToddTrimble: But obviously he just meant "complex conjugation" as a shorthand for the $\zeta \mapsto \zeta^{-1}$ automorphism of K/L (as in MZ's answer) just like he meant the norm to be defined using the obvious reinterpretation which doesn't refer to complex numbers (rather than the definition he actually used). | |
| Aug 23, 2013 at 17:15 | comment | added | Makoto Kato |
@MichaelZieve [@quid: of course you can define positivity in many ways. I was trying to get MK to give a definition, since doing so would indicate what type of proof he sought. Let me add that I don't understand the upvotes for MK's definition that an integer is positive if it is a positive integer.] I said an integer is positive if and only if it is a natural number. Could you explain why this definition does not make sense if that is what you are claiming?
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| Aug 23, 2013 at 16:36 | comment | added | Yemon Choi | @ToddTrimble OK, in that case I agree with your reading (on both counts). My point was: if MK is dissatisfied with the "obvious" interpretation of his question, then maybe, just maybe, it wasn't that well posed in the first place. (I like my circulant formulation better, but then I'm far from unbiased.) I also find the attitude displayed by "no one wanted to answer my question on MSE, so it must be research-level in the sense of MO", rather trying - see an earlier version of this meta.MO question which invoked FLT | |
| Aug 23, 2013 at 16:32 | comment | added | Todd Trimble | @YemonChoi It's not obvious to me why Felipe's argument wouldn't count as "purely algebraic", but it's possible that MK doesn't think it's pure enough since the theory of real closed fields uses Zorn's lemma at places, whereas this result ought to be explicable purely in terms of finite-dimensional extensions of $\mathbb{Q}$. | |
| Aug 23, 2013 at 16:22 | comment | added | Yemon Choi | MK has now left a comment on the MO question, saying that as far as he is concerned " A real closure of Q is isomorphic to the field of real algebraic numbers. Hence the proof using it is basically the same as stated in the question." So would those who think it is "obvious" what the OP meant in his question, but who think that Felipe Voloch's argument answers the question, please enlighten me as to what is going on? | |
| Aug 23, 2013 at 16:17 | comment | added | Todd Trimble | Noah, the original argument of Kato's did refer to complex conjugation (and hence the real numbers as fixed field). Michael's answer fleshed out what the OP needed, which involves precisely the fact that the fixed field of the automorphism that sends $\zeta$ to $\zeta^{-1}$ is formally real; for that one needs a total ordering on the fixed field. Your answer says nothing about any of that, and IMO was not helpful. | |
| Aug 23, 2013 at 15:45 | comment | added | Noah Snyder | But @ToddTrimble, Michael Zieve's answer at m.SE is saying exactly the same thing as my answer here: the original argument doesn't "use the real numbers." | |
| Aug 23, 2013 at 14:44 | comment | added | Michael Zieve | Fair enough. Sorry I got carried away. | |
| Aug 23, 2013 at 13:49 | comment | added | Todd Trimble | Oh, for heaven's sakes. Of course the question makes sense, because there is just one way to make the rationals into an ordered field. This fact is independent of whether or not the OP knows the formal details (and I think it would be more productive not to badger the OP over these, since he/she might not know them, but to help him/her by incorporating them into an answer). I am glad Michael Zieve gave a productive answer at MSE. | |
| Aug 23, 2013 at 12:30 | comment | added | user9072 | @MichaelZieve I only saw your other comment after having written mine. Regarding MK's definition I think it depends how one starts; as you know, a usual construction of the integers is as equivalence classes of pairs of natural numbers. And than one imbeds the natural numbers. Then the natural number so to say 'existed' before the integers and the definition is not circular. (Or one could also say as usual 2 is shorthand for 1 + 1 and so on, then it also makes sense the positive one are thos that can be written as sums of 1, the mult identity of the domain 'integers') | |
| Aug 23, 2013 at 12:25 | comment | added | Michael Zieve | @quid: of course you can define positivity in many ways. I was trying to get MK to give a definition, since doing so would indicate what type of proof he sought. Let me add that I don't understand the upvotes for MK's definition that an integer is positive if it is a positive integer. | |
| Aug 23, 2013 at 12:01 | comment | added | user9072 | And if one wants to have something more formal one could define the positive integers as those integers that can be written as the sum of four squares of integers (not all of them 0). | |
| Aug 23, 2013 at 3:21 | comment | added | Makoto Kato |
@MichaelZieve [And how do you define positivity of an integer?] Natural numbers $1, 2, 3,\dots$ are exactly (strictly) positive integers.
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| Aug 23, 2013 at 3:15 | comment | added | Michael Zieve | And how do you define positivity of an integer? | |
| Aug 23, 2013 at 3:01 | comment | added | Makoto Kato |
@MichaelZieve [@MakotoKato: Noah is saying the question doesn't make sense because you didn't define what it means for a rational number to be positive. Of course, everyone knows what it means for a real number to be positive, but the point of your question is to avoid mentioning real numbers. So you need to say what definition you have in mind for positive rationals.] Let $r = a/b$ be a rational number where $a$ and $b$ are rational integers. If $a \gt 0$ and $b\gt 0$, we say $r$ is positive. Regards,
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| Aug 23, 2013 at 1:21 | comment | added | Michael Zieve | @MakotoKato: Noah is saying the question doesn't make sense because you didn't define what it means for a rational number to be positive. Of course, everyone knows what it means for a real number to be positive, but the point of your question is to avoid mentioning real numbers. So you need to say what definition you have in mind for positive rationals. Also, Noah didn't say that you know the answer to your question, he said that you know a proof of the result in question. | |
| Aug 22, 2013 at 23:30 | comment | added | Makoto Kato | The question makes sense. Please read my last comment to Yemon Choi in the thread. No, I don't know the answer to my question. If you know it, could you answer it in MSE? | |
| Aug 22, 2013 at 23:19 | history | answered | Noah Snyder | CC BY-SA 3.0 |