File:Discriminant49CubicFieldFundamentalDomainOfUnits.png
Jump to navigation
Jump to search
Size of this preview: 800 × 468 pixels. Other resolutions: 320 × 187 pixels | 640 × 375 pixels | 839 × 491 pixels.
Original file (839 × 491 pixels, file size: 38 KB, MIME type: image/png)
Captions
Captions
Add a one-line explanation of what this file represents
Summary
[edit]
| Description |
English: A fundamental domain in logarithmic space of the group of units of the number field K obtained by adjoining to Q a root of f(x) = x3 + x2 − 2x − 1. This field is a Galois extension of degree 3. Its discriminant is 49 = 72. If α denotes the root of f(x) which is approximately 1.24698, then a set of fundamental units is {ε1, ε2} where ε1 = α2 + α − 1 and ε2 = 2 − α2. The fundamental domain is spanned by v1 and v2 with the ith component of vj equal to log|σi(εj)|. So, approximately, v1 = (0.588863, −0.809587, 0.220724) and v1 = (−0.809587, 0.220724, 0.588863). The area of the fundamental domain is approximately 0.910114, so the regulator of K is approximately 0.525455. See Table B.4 of Cohen's A course in computational algebraic number theory (1993). Computed using PARI/GP. Plotted using Mathematica. |
| Date | |
| Source | Own work |
| Author | RobHar |
This diagram was created with Mathematica by n.
Licensing
[edit]
I, the copyright holder of this work, hereby publish it under the following licenses:
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
|
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. |
You may select the license of your choice.
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 00:39, 14 June 2010 | | 839 × 491 (38 KB) | RobHar (talk | contribs) | {{Information |Description={{en|1=A fundamental domain in logarithmic space of the group of units of the number field ''K'' obtained by adjoining to '''Q''' a root of ''f''(''x'') = ''x''<sup>3</sup> + ''x''<sup>2</sup> −&nb |
You cannot overwrite this file.
File usage on Commons
There are no pages that use this file.
File usage on other wikis
The following other wikis use this file:
- Usage on en.wikipedia.org
- Usage on es.wikipedia.org
- Usage on ja.wikipedia.org
