Editing Dirichlet's unit theorem (section)

{{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}}