Timeline for Characterizing all simple algebraic ring extensions of $\mathbb{C}[x]$ having no prime elements
Current License: CC BY-SA 3.0
11 events
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| May 12, 2017 at 12:59 | comment | added | user237522 | @user26857, I guess that your comment is closely related to the second answer in math.stackexchange.com/questions/2277193/…. | |
| May 12, 2017 at 0:37 | comment | added | Mohan | Just as an example, consider $R=\mathbb{C}[x][w]$ with $w^2=x^4+x$. You can easily check that this is a smooth curve of genus one. But, $w-x^2$ is a prime defining the origin $x=w=0$. | |
| May 12, 2017 at 0:00 | comment | added | Mohan | I would suspect the integral case is simpler, but I do not think there can be any simple criterion without dealing with what happens at infinity. | |
| May 11, 2017 at 22:09 | comment | added | user237522 | @Mohan, please, do you have any ideas what happens if $f$ is not monic, namely, $w$ is not integral? Which case (integral or not integral) is easier to deal with, in your opinion? | |
| May 11, 2017 at 21:57 | comment | added | Mohan | @MooS This is not quite correct, several things can go wrong. In particular, there could be several points at infinity and then things don't work so easily. | |
| May 11, 2017 at 18:55 | comment | added | user237522 | Thank you very much! Additional partial answers (from you or other people) are also welcome. | |
| May 11, 2017 at 18:14 | comment | added | MooS | This is far from being a full characterisation: In the integral case, if the curve given by $f$ is smooth, then there are no prime elements whenever the curve has positive genus. Because a prime element would mean that the corresponding point on the curve is linear equivalent to the point at infinity in the projective closure. This can only happen for rational curves by a well known lemma. | |
| May 11, 2017 at 14:54 | history | edited | user237522 | CC BY-SA 3.0 |
added 27 characters in body
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| S May 11, 2017 at 14:52 | history | suggested | David Richter | CC BY-SA 3.0 |
the title had mismatched braces i fixed them
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| May 11, 2017 at 14:51 | review | Suggested edits | |||
| S May 11, 2017 at 14:52 | |||||
| May 11, 2017 at 14:48 | history | asked | user237522 | CC BY-SA 3.0 |