Editing Dirichlet's unit theorem

{{short description|Gives the rank of the group of units in the ring of algebraic integers of a number field}}
In [[mathematics]], '''Dirichlet's unit theorem''' is a basic result in [[algebraic number theory]] due to [[Peter Gustav Lejeune Dirichlet]].<ref>{{harvnb|Elstrodt|2007|loc=§8.D}}</ref> It determines the [[rank of an abelian group|rank]] of the [[group of units]] in the [[ring (mathematics)|ring]] {{math|''O''<sub>''K''</sub>}} of [[algebraic integer]]s of a [[number field]] {{mvar|K}}. The '''regulator''' is a positive real number that determines how "dense" the units are.

The statement is that the group of units is finitely generated and has [[Rank of an abelian group|rank]] (maximal number of multiplicatively independent elements) equal to
{{block indent|em=1.5|text={{math|1=''r'' = ''r''<sub>1</sub> + ''r''<sub>2</sub> − 1}}}}
where {{math|''r''<sub>1</sub>}} is the ''number of real embeddings'' and {{math|''r''<sub>2</sub>}} the ''number of conjugate pairs of complex embeddings'' of {{mvar|K}}. This characterisation of {{math|''r''<sub>1</sub>}} and {{math|''r''<sub>2</sub>}} is based on the idea that there will be as many ways to embed {{mvar|K}} in the [[complex number]] field as the degree <math>n = [K: \mathbb{Q}]</math>; these will either be into the [[real number]]s, or pairs of embeddings related by [[complex conjugation]], so that
{{block indent|em=1.5|text={{math|1=''n'' = ''r''<sub>1</sub> + 2''r''<sub>2</sub>}}.}}

Note that if {{mvar|K}} is [[Galois extension|Galois]] over <math>\mathbb{Q}</math> then either {{math|1=''r''<sub>1</sub> = 0}} or {{math|1=''r''<sub>2</sub> = 0}}.

Other ways of determining {{math|''r''<sub>1</sub>}} and {{math|''r''<sub>2</sub>}} are

* use the [[Primitive element (field theory)|primitive element]] theorem to write <math>K = \mathbb{Q}(\alpha)</math>, and then {{math|''r''<sub>1</sub>}} is the number of [[conjugate element (field theory)|conjugates]] of {{mvar|α}} that are real, {{math|2''r''<sub>2</sub>}} the number that are complex; in other words, if {{mvar|''f''}} is the minimal polynomial of {{mvar|α}} over <math>\mathbb{Q}</math>, then {{math|''r''<sub>1</sub>}} is the number of real roots and {{math|''2r''<sub>2</sub>}} is the number of non-real complex roots of {{mvar|''f''}} (which come in complex conjugate pairs);
* write the [[tensor product of fields]] <math>K \otimes_{\mathbb{Q}} \mathbb{R}</math> as a product of fields, there being {{math|''r''<sub>1</sub>}} copies of <math>\mathbb{R}</math> and {{math|''r''<sub>2</sub>}} copies of <math>\mathbb{C}</math>.

As an example, if {{mvar|K}} is a [[quadratic field]], the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. The theory for real quadratic fields is essentially the theory of [[Pell's equation]].

The rank is positive for all number fields besides <math>\mathbb{Q}</math> and imaginary quadratic fields, which have rank 0. The 'size' of the units is measured in general by a [[determinant]] called the regulator. In principle a basis for the units can be effectively computed; in practice the calculations are quite involved when {{mvar|n}} is large.

The torsion in the group of units is the set of all roots of unity of {{mvar|K}}, which form a finite [[cyclic group]]. For a number field with at least one real embedding the torsion must therefore be only {{math|{1,−1{{)}}}}. There are number fields, for example most [[imaginary quadratic field]]s, having no real embeddings which also have {{math|{1,−1{{)}}}} for the torsion of its unit group.

Totally real fields are special with respect to units. If {{math|''L''/''K''}} is a finite extension of number fields with degree greater than 1 and
the units groups for the integers of {{mvar|L}} and {{mvar|K}} have the same rank then {{mvar|K}} is totally real and {{mvar|L}} is a totally complex quadratic extension. The converse holds too. (An example is {{mvar|K}} equal to the rationals and {{mvar|L}} equal to an imaginary quadratic field; both have unit rank 0.)

The theorem not only applies to the maximal order {{mvar|O<sub>K</sub>}} but to any order {{math|''O'' ⊂ ''O''<sub>K</sub>}}.<ref>{{cite book|title=Number Rings|first=P.|last=Stevenhagen|year=2012|url=http://websites.math.leidenuniv.nl/algebra/ant.pdf| page=57}}</ref>

There is a generalisation of the unit theorem by [[Helmut Hasse]] (and later [[Claude Chevalley]]) to describe the structure of the group of ''[[S-unit|{{mvar|S}}-unit]]s'', determining the rank of the unit group in [[localization of a ring|localizations]] of rings of integers. Also, the [[Galois module]] structure of <math>\mathbb{Q} \oplus O_{K, S} \otimes_{\mathbb{Z}} \mathbb{Q}</math> has been determined.{{sfn|Neukirch|Schmidt|Wingberg|2000|loc=proposition VIII.8.6.11}}

==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>

==Higher regulators==

A 'higher' regulator refers to a construction for a function on an [[algebraic K-group|algebraic {{mvar|K}}-group]] with index {{math|''n'' > 1}} that plays the same role as the classical regulator does for the group of units, which is a group {{math|''K''<sub>1</sub>}}. A theory of such regulators has been in development, with work of [[Armand Borel]] and others. Such higher regulators play a role, for example, in the [[Beilinson conjectures]], and are expected to occur in evaluations of certain [[L-function|{{mvar|L}}-function]]s at integer values of the argument.<ref name=Bloch>{{cite book | last=Bloch | first=Spencer J. | author-link=Spencer Bloch | title=Higher regulators, algebraic {{mvar|K}}-theory, and zeta functions of elliptic curves | series=CRM Monograph Series | volume=11 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2000 | isbn=0-8218-2114-8 | zbl=0958.19001 }}</ref> See also [[Beilinson regulator]].

==Stark regulator==
The formulation of [[Stark's conjectures]] led [[Harold Stark]] to define what is now called the '''Stark regulator''', similar to the classical regulator as a determinant of logarithms of units, attached to any [[Artin representation]].<ref>{{cite report|title=A Report on Artin's holomorphy conjecture| first1=Dipendra| last1=Prasad| first2=C. S. |last2=Yogonanda |date=2007-02-23 |url=http://www.math.tifr.res.in/~dprasad/artin.pdf}}</ref><ref>{{cite thesis|title=Stark's Conjectures |first=Samit |last=Dasgupta |year=1999 |url=http://www.math.harvard.edu/~dasgupta/papers/Dasguptaseniorthesis.pdf |url-status=dead |archive-url=https://web.archive.org/web/20080510150747/http://www.math.harvard.edu/~dasgupta/papers/Dasguptaseniorthesis.pdf |archive-date=2008-05-10 }}</ref>

=={{mvar|p}}-adic regulator==
Let {{mvar|K}} be a [[number field]] and for each [[Valuation (algebra)|prime]] {{mvar|P}} of {{mvar|K}} above some fixed rational prime {{mvar|p}}, let {{math|''U''<sub>''P''</sub>}} denote the local units at {{mvar|P}} and let {{math|''U''<sub>1,''P''</sub>}} denote the subgroup of principal units in {{math|''U''<sub>''P''</sub>}}. Set <math display="block"> U_1 = \prod_{P|p} U_{1,P}. </math>

Then let {{math|''E''<sub>1</sub>}} denote the set of global units {{mvar|ε}} that map to {{math|''U''<sub>1</sub>}} via the diagonal embedding of the global units in {{mvar|E}}.

Since {{math|''E''<sub>1</sub>}} is a finite-[[Index of a subgroup|index]] subgroup of the global units, it is an [[abelian group]] of rank {{math|''r''<sub>1</sub> + ''r''<sub>2</sub> − 1}}. The '''{{mvar|p}}-adic regulator''' is the determinant of the matrix formed by the {{mvar|p}}-adic logarithms of the generators of this group. ''[[Leopoldt's conjecture]]'' states that this determinant is non-zero.<ref name=NSW6267>Neukirch et al. (2008) p. 626–627</ref><ref>{{cite book | last=Iwasawa | first=Kenkichi | author-link=Kenkichi Iwasawa | title=Lectures on {{mvar|p}}-adic {{mvar|L}}-functions | series=Annals of Mathematics Studies | volume=74 | location=Princeton, NJ | publisher=Princeton University Press and University of Tokyo Press | year=1972 | isbn=0-691-08112-3 | zbl=0236.12001 | pages=36–42 }}</ref>

==See also==
*[[Elliptic unit]]
*[[Cyclotomic unit]]
*[[Shintani's unit theorem]]

==Notes==
{{Reflist}}

==References==
* {{Cite book | last=Cohen | first=Henri | author-link=Henri Cohen (number theorist) | title=A Course in Computational Algebraic Number Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | volume=138 | isbn=978-3-540-55640-4 | mr=1228206 | zbl=0786.11071 | year=1993 }}
*{{cite journal | last = Elstrodt | first = Jürgen | journal = Clay Mathematics Proceedings | title = The Life and Work of Gustav Lejeune Dirichlet (1805–1859) | year = 2007 | url = http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | access-date = 2010-06-13 | archive-date = 2021-05-22 | archive-url = https://web.archive.org/web/20210522140235/https://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | url-status = dead }}
* {{cite book | first=Serge | last=Lang | author-link=Serge Lang | title=Algebraic number theory | edition=2nd | series=Graduate Texts in Mathematics | volume=110 | location=New York | publisher=[[Springer-Verlag]] | year=1994 | isbn=0-387-94225-4 | zbl=0811.11001 }}
*{{Neukirch ANT}}
*{{Neukirch et al. CNF}}

[[Category:Theorems in algebraic number theory]]