Timeline for Fundamental units of imaginary quartic fields
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 23, 2013 at 13:24 | vote | accept | Jean Raimbault | ||
| Jul 23, 2013 at 11:44 | comment | added | Noam D. Elkies | One must also consider Pell equations $t^2 - u^2 d = -4$ (and also $\pm 4i$ and $\pm 2 \pm \sqrt{-12}$ for $D=-4$ and $D=-3$ respectively). But these do not materially affect the answer. | |
| Jul 23, 2013 at 11:42 | answer | added | Noam D. Elkies | timeline score: 8 | |
| Jul 23, 2013 at 7:45 | history | edited | Jean Raimbault | CC BY-SA 3.0 |
deleted 7 characters in body; edited title
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| Jul 23, 2013 at 7:44 | comment | added | Jean Raimbault | @KConrad: quartic it is, then. | |
| Jul 23, 2013 at 7:43 | comment | added | Jean Raimbault | @anton: yes, $|\varepsilon_F|$ depends only on $d$; $D$ determines the range of values of $d$. | |
| Jul 23, 2013 at 2:56 | comment | added | KConrad | I think it might be better to call these fields quartic rather than biquadratic: quartic definitely means "degree 4", while biquadratic suggests a specific type of quartic extension, namely a composite of two quadratic extensions (so of the form ${\mathbf Q}(\sqrt{a},\sqrt{b})$ with rational $a$ and $b$). | |
| Jul 22, 2013 at 16:39 | comment | added | user1688 | Yes, but then $\varepsilon_F$ depends only on $d$ and not on $D$? | |
| Jul 22, 2013 at 15:38 | comment | added | Jean Raimbault | I believe that "fundamental discriminant" is the standard term for the set of these integers which are quadratic residues mod 4 but not squares in ${\mathcal O}_D$ (I added "in ${\mathcal O}_D$" to put emphasis on the dependancy on $D$). | |
| Jul 22, 2013 at 14:40 | comment | added | user1688 | What do you mean by "fundamental discriminant"? Usually, it's an integer, so that would mean that always all fd are in ${\cal O}_D$. | |
| Jul 22, 2013 at 13:57 | history | asked | Jean Raimbault | CC BY-SA 3.0 |