Timeline for On the partner of the Emma Lehmer quintic
Current License: CC BY-SA 3.0
12 events
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| Nov 23, 2016 at 15:58 | comment | added | Tito Piezas III | @NoamD.Elkies: Ok. Ultimately, my interest at inspecting the coefficients of certain quintics as discussed in the other post is to find another cyclic family similar to Lehmer's. This 2005 paper gives a $7$th deg version, but frustratingly does not provide an explicit second quintic family. | |
| Nov 23, 2016 at 15:49 | comment | added | Noam D. Elkies | I think you can easily get examples that don't satisfy (1) by starting from any cyclic quintic field and applying a random Tschirnhauser transformation (i.e. choosing some other generator of the field). Conversely, the argument I gave using quadratic forms shows that there's always some generator that does satisfy (1). | |
| Nov 23, 2016 at 15:04 | comment | added | Tito Piezas III | @NoamD.Elkies: Thanks. I found this cyclic quintic $x^5 + 2x^4 - 665x^3 + 5289x^2 + 6835x - 1453=0$ in Kluener's database and it does not obey (1). However, it seems the quintics for the family $\cos(\pi/p)$ and prime $p\equiv 1\pmod{10}$ do obey it and some Diophantine relations. Kindly see this MO post. | |
| Nov 22, 2016 at 2:31 | comment | added | Noam D. Elkies | Certainly not, because you can choose any generator of the field. But you can still ask whether every cyclic quintic field $F/{\bf Q}$ has some generator whose Galois orbit satisfies that equation, and there the answer is Yes: a quadratic form in at least $5$ variables has nontrivial rational zeros if and only if it has nontrivial real zeros, and this single condition is readily checked for the form (1). | |
| Nov 22, 2016 at 1:36 | comment | added | Tito Piezas III | @NoamD.Elkies: While testing this MSE list, is it true that all cyclic quintics obey $(1)$? | |
| Nov 22, 2016 at 1:33 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
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| Nov 21, 2016 at 4:46 | vote | accept | Tito Piezas III | ||
| Nov 21, 2016 at 4:38 | comment | added | Noam D. Elkies | As written the answer to the "Q" is clearly "Yes", but the subset might turn out (and as it happens does turn out) to be empty :-) | |
| Nov 21, 2016 at 2:11 | answer | added | Noam D. Elkies | timeline score: 8 | |
| Nov 20, 2016 at 15:51 | history | edited | GH from MO |
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| Nov 20, 2016 at 8:19 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
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| Nov 20, 2016 at 8:11 | history | asked | Tito Piezas III | CC BY-SA 3.0 |