Timeline for Extensions of $\mathbb{Z}[\sqrt{-n}]$ that are UFD
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2017 at 14:39 | comment | added | paul garrett | Staying within the ambient field, if you localize by allowing denominators in finitely many prime ideals representing the ideal classes, the resulting slightly-localized ring is always a PID, for example. This is an old, standard result. | |
| Sep 12, 2017 at 14:32 | comment | added | Martin | @KConrad I edited the question | |
| Sep 12, 2017 at 14:31 | history | edited | Martin | CC BY-SA 3.0 |
clarified question
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| Sep 12, 2017 at 13:48 | comment | added | KConrad | This question is waaaay too vague to have a useful answer. Every field and DVR (e.g., localizing the ring of integers of $\mathbf Q(\sqrt{-n})$ at a prime ideal) is a UFD. A field in turn has a ton of its own field extensions, and all are UFDs. It does not seem very interesting. Be more focused on explaining what you are really looking for and why. | |
| Sep 12, 2017 at 13:20 | comment | added | Martin | @Wojowu yes (preferably "small"), but first I won't pose any condition other then it should be a ring containing $\mathbb{Z}[\sqrt{-n}]$ | |
| Sep 12, 2017 at 12:57 | comment | added | Wojowu | What do you mean by "extension"? Just a ring which contains $\mathbb Z[\sqrt{-n}]$? | |
| Sep 12, 2017 at 12:50 | history | asked | Martin | CC BY-SA 3.0 |