Timeline for Compute polynomial $p(x)$ if $x^5=1,\, x\neq 1$ [reducing mod $\textit{simpler}$ multiples]
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 27, 2019 at 12:04 | audit | First posts | |||
| May 27, 2019 at 12:32 | |||||
| May 19, 2019 at 18:17 | audit | First posts | |||
| May 19, 2019 at 18:20 | |||||
| May 18, 2019 at 21:16 | comment | added | LSpice | Note that, although of course this is the cyclotomic polynomial $\Phi_5(x)$, you need not know cyclotomic polynomials to think of this; a good high-school student should be able to sum (finite) geometric series, of which $Q(x)$ is one. (For example, one can easily use a similar trick for $x^5 + \dotsb + x + 1$, even though that is not $\Phi_6(x)$.) | |
| May 16, 2019 at 3:04 | audit | First posts | |||
| May 16, 2019 at 3:05 | |||||
| May 16, 2019 at 1:14 | comment | added | Rushabh Mehta | @not2qubit That's fascinating. I'm aware that the term cyclotomic polynomial isn't used everywhere, but nearly every recent STEM graduate I've met can recognize them (at least prime cyclotomic polynomials) when they see them. Maybe I'm experiencing selection bias... | |
| May 15, 2019 at 18:00 | comment | added | not2qubit | I think I'm getting old... I have PhD in the theoretical sciences and I have never heard of cyclotomic polynomials until now! This is absolutely not obvious, and surely not intuitive in any way, unless you have actually been lectured on it. So I'm also happy to see how much HS math education have improved. | |
| May 14, 2019 at 23:53 | comment | added | Bill Dubuque | @FedericoPoloni I agree that this is likely in the standard bag of tricks for competitions. But as I stress in my answer, the idea behind this is already known at early levels, and will usually be explicitly recognized when one studies university level algebra (if not earlier). I don't recall seeing it ever explicitly mentioned in textbooks. | |
| May 14, 2019 at 17:09 | comment | added | Federico Poloni | @MooseBoys I agree with your estimate. The question is "is this obvious?", and my comment is "it is a standard problem for people who train for math Olympiads". I realize that it's a strong additional assumption. I'm not arguing that it's a good or bad problem: as you point out, to do that, we would at least have to know to which students it was given. | |
| May 14, 2019 at 17:01 | comment | added | MooseBoys | @Federico Where in the question does it say that this is an Olympiad test? OP phrased their question as if it were applicable to all high school students. I'd guess that fewer than 5% of all students would get the correct answer. | |
| May 14, 2019 at 9:12 | comment | added | Federico Poloni | It may not be obvious in other contexts since you never had to use it, but I assure you that in Math Olympiad training it's a standard trick. All serious contestants will probably already have seen something similar. There's nothing outrageous or shameful in that; Olympiad-style math is simply different from what people use and need in other professional contexts (physicist/engineer) --- the focus on Euclidean geometry being a perfect example. | |
| May 14, 2019 at 9:04 | comment | added | WoJ | @MooseBoys: I would go even further. As a PhD in Physics (admittedly many years ago), this is not at all obvious to me. | |
| May 14, 2019 at 6:58 | comment | added | MooseBoys | I would go even further. As a BSE graduate in engineering, this is not at all obvious to me. | |
| May 13, 2019 at 18:58 | history | answered | DanLewis3264 | CC BY-SA 4.0 |