Timeline for Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 15, 2023 at 21:01 | vote | accept | user108580 | ||
| Nov 10, 2023 at 19:05 | answer | added | user516477 | timeline score: 4 | |
| Nov 9, 2023 at 12:32 | comment | added | Zach Teitler | Both $x$ and $x^3+y^3+1$ vanish at the point $(0,1)$ hence one of the factors also must vanish there. Perhaps it's easier to expand polynomials in a Taylor series around $(0,1)$ instead of the origin? | |
| Nov 9, 2023 at 9:22 | comment | added | Balazs | ...I have also been pondering the fact that the real story diverges: ${\mathbb R}[x,y]/(x^2+y^2-1)$ is not a UFD, the real curve has a non-trivial class group, but $\{x^2+y^2=1\}\subset{\mathbb R}^2$ still admits a rational parametrisation (indeed any base field works), since it has obvious real points. | |
| Nov 9, 2023 at 9:20 | comment | added | Balazs | ...what appears very interesting to me is that the question of the rings being UFD should somehow be related to whether the corresponding curves have rational parametrisations: $\{x^2+y^2=1\}\subset{\mathbb C}^2$ of course does (elementary, project from any point), whereas $\{x^3+y^3=1\}\subset{\mathbb C}^2$ does not; the latter also admits an elementary proof (involving infinite descent). This is the other well-known shadow of the effect of genus. | |
| Nov 9, 2023 at 9:16 | comment | added | Balazs | I have been wondering about this myself, especially in comparison with the related example of the ring ${\mathbb C}[x,y]/(x^2+y^2-1)$, which is a UFD; this has an elementary proof, but also another proof that studies the Picard group of the corresponding punctured curve. | |
| Nov 7, 2023 at 18:16 | comment | added | Will Chen | I'm curious if there is a relatively clean proof that doesn't involve knowing that this is the coordinate ring of a positive genus curve. | |
| Nov 7, 2023 at 16:40 | history | edited | GH from MO |
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| Nov 7, 2023 at 16:40 | answer | added | GH from MO | timeline score: 12 | |
| Nov 7, 2023 at 5:27 | comment | added | abx | Yes, if you know a bit of algebraic geometry: you are asking about the Picard group of an elliptic curve minus 3 points — this is indeed very large. | |
| Nov 7, 2023 at 5:11 | comment | added | KReiser | Please see here on meta for discussion about cross-posting between MO and MSE. (Here is a link to the MSE post.) | |
| S Nov 7, 2023 at 4:47 | review | First questions | |||
| Nov 7, 2023 at 7:05 | |||||
| S Nov 7, 2023 at 4:47 | history | asked | user108580 | CC BY-SA 4.0 |