Editing Dirichlet's unit theorem (section)

==The regulator==

Suppose that ''K'' is a number field and <math>u_1, \dots, u_r</math> are a set of generators for the unit group of ''K'' modulo roots of unity. There will be {{math|''r'' + 1}} Archimedean places of ''K'', either real or complex. For <math>u\in K</math>, write <math>u^{(1)},\dots,u^{(r+1)}</math> for the different embeddings into <math>\mathbb{R}</math> or <math>\mathbb{C}</math> and set {{math|''N''<sub>''j''</sub>}} to 1 or 2 if the corresponding embedding is real or complex respectively. Then the {{math|''r'' × (''r'' + 1)}} matrix <math display="block">\left(N_j\log \left|u_i^{(j)}\right|\right)_{i=1,\dots,r,\; j=1,\dots,r+1}</math> has the property that the sum of any row is zero (because all units have norm 1, and the log of the norm is the sum of the entries in a row). This implies that the absolute value {{mvar|R}} of the determinant of the submatrix formed by deleting one column is independent of the column. The number {{mvar|R}} is called the '''regulator''' of the algebraic number field (it does not depend on the choice of generators {{math|''u''<sub>''i''</sub>}}). It measures the "density" of the units: if the regulator is small, this means that there are "lots" of units.

The regulator has the following geometric interpretation. The map taking a unit {{mvar|u}} to the vector with entries <math display="inline">N_j\log \left|u^{(j)}\right|</math> has an image in the {{mvar|r}}-dimensional subspace of <math>\mathbb{R}^{r + 1}</math> consisting of all vectors whose entries have sum 0, and by Dirichlet's unit theorem the image is a lattice in this subspace. The volume of a fundamental domain of this lattice is <math>R\sqrt{r + 1}</math>.

The regulator of an algebraic number field of degree greater than 2 is usually quite cumbersome to calculate, though there are now computer algebra packages that can do it in many cases. It is usually much easier to calculate the product {{math|''hR''}} of the [[class number (number theory)|class number]] {{mvar|h}} and the regulator using the [[class number formula]], and the main difficulty in calculating the class number of an algebraic number field is usually the calculation of the regulator.

===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>