Editing Dirichlet's unit theorem (section)

===Examples===
[[Image:Discriminant49CubicFieldFundamentalDomainOfUnits.png|thumb|300px|right|A fundamental domain in logarithmic space of the group of units of the cyclic cubic field {{mvar|K}} obtained by adjoining to <math>\mathbb{Q}</math> a root of {{math|1=''f''(''x'') = ''x''<sup>3</sup> + ''x''<sup>2</sup> − 2''x'' − 1}}. If {{mvar|α}} denotes a root of {{math|''f''(''x'')}}, then a set of fundamental units is {{math|{''ε''<sub>1</sub>, ''ε''<sub>2</sub>}<nowiki/>}}, where {{math|1=''ε''<sub>1</sub> = ''α''<sup>2</sup> + ''α'' − 1}} and {{math|1=''ε''<sub>2</sub> = 2 − ''α''<sup>2</sup>}}. The area of the fundamental domain is approximately 0.910114, so the regulator of {{mvar|K}} is approximately 0.525455.]]
*The regulator of an [[imaginary quadratic field]], or of the rational integers, is 1 (as the determinant of a {{math|0 × 0}} matrix is 1).
*The regulator of a [[real quadratic field]] is the logarithm of its [[Fundamental unit (number theory)|fundamental unit]]: for example, that of <math>\mathbb{Q}(\sqrt{5})</math> is <math display="inline">\log \frac{\sqrt{5} + 1}{2}</math>. This can be seen as follows. A fundamental unit is <math display="inline">(\sqrt{5} + 1) / 2</math>, and its images under the two embeddings into <math>\mathbb{R}</math> are <math display="inline">(\sqrt{5} + 1) / 2</math> and <math display="inline">(-\sqrt{5} + 1) / 2</math>. So the {{math|''r'' × (''r'' + 1)}} matrix is <math display="block">\left[1\times\log\left|\frac{\sqrt{5} + 1}{2}\right|, \quad 1\times \log\left|\frac{-\sqrt{5} + 1}{2}\right|\ \right].</math>
*The regulator of the [[cyclic cubic field]] <math>\mathbb{Q}(\alpha)</math>, where {{mvar|α}} is a root of {{math|''x''<sup>3</sup> + ''x''<sup>2</sup> − 2''x'' − 1}}, is approximately 0.5255. A basis of the group of units modulo roots of unity is {{math|{''ε''<sub>1</sub>, ''ε''<sub>2</sub>}<nowiki/>}} where {{math|1=''ε''<sub>1</sub> = ''α''<sup>2</sup> + ''α'' − 1}} and {{math|1=''ε''<sub>2</sub> = 2 − ''α''<sup>2</sup>}}.<ref>{{harvnb|Cohen|1993|loc=Table B.4}}</ref>