File:Discriminant49CubicFieldFundamentalDomainOfUnits.png
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Summary
| 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.
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 00:39, 14 June 2010 | | 839 × 491 (38 KB) | RobHar | {{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 |
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