# Dirichlet's unit theorem

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Dirichlet's_unit_theorem
> Markdown URL: https://mediated.wiki/source/Dirichlet's_unit_theorem.md
> Source: https://en.wikipedia.org/wiki/Dirichlet's_unit_theorem
> Source revision: 1321322368
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [mathematics](/source/Mathematics), **Dirichlet's unit theorem** is a basic result in [algebraic number theory](/source/Algebraic_number_theory) due to [Peter Gustav Lejeune Dirichlet](/source/Peter_Gustav_Lejeune_Dirichlet).[1] It determines the [rank](/source/Rank_of_an_abelian_group) of the [group of units](/source/Group_of_units) in the [ring](/source/Ring_(mathematics)) *O**K* of [algebraic integers](/source/Algebraic_integer) of a [number field](/source/Number_field) 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](/source/Rank_of_an_abelian_group) (maximal number of multiplicatively independent elements) equal to

*r* = *r*1 + *r*2 − 1

where *r*1 is the *number of real embeddings* and *r*2 the *number of conjugate pairs of complex embeddings* of K. This characterisation of *r*1 and *r*2 is based on the idea that there will be as many ways to embed K in the [complex number](/source/Complex_number) field as the degree n = [K: \mathbb{Q}]; these will either be into the [real numbers](/source/Real_number), or pairs of embeddings related by [complex conjugation](/source/Complex_conjugation), so that

*n* = *r*1 + 2*r*2.

Note that if K is [Galois](/source/Galois_extension) over \mathbb{Q} then either *r*1 = 0 or *r*2 = 0.

Other ways of determining *r*1 and *r*2 are

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

As an example, if K is a [quadratic field](/source/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](/source/Pell's_equation).

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

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

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

The theorem not only applies to the maximal order OK but to any order *O* ⊂ *O*K.[2]

There is a generalisation of the unit theorem by [Helmut Hasse](/source/Helmut_Hasse) (and later [Claude Chevalley](/source/Claude_Chevalley)) to describe the structure of the group of *[S-units](/source/S-unit)*, determining the rank of the unit group in [localizations](/source/Localization_of_a_ring) of rings of integers. Also, the [Galois module](/source/Galois_module) structure of \mathbb{Q} \oplus O_{K, S} \otimes_{\mathbb{Z}} \mathbb{Q} has been determined.[3]

## The regulator

Suppose that *K* is a number field and u_1, \dots, u_r are a set of generators for the unit group of *K* modulo roots of unity. There will be *r* + 1 Archimedean places of *K*, either real or complex. For u\in K, write u^{(1)},\dots,u^{(r+1)} for the different embeddings into \mathbb{R} or \mathbb{C} and set *N**j* to 1 or 2 if the corresponding embedding is real or complex respectively. Then the *r* × (*r* + 1) matrix

\left(N_j\log \left|u_i^{(j)}\right|\right)_{i=1,\dots,r,\; j=1,\dots,r+1}

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 R of the determinant of the submatrix formed by deleting one column is independent of the column. The number R is called the **regulator** of the algebraic number field (it does not depend on the choice of generators *u**i*). 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 u to the vector with entries N_j\log \left|u^{(j)}\right| has an image in the r-dimensional subspace of \mathbb{R}^{r + 1} 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 R\sqrt{r + 1}.

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 *hR* of the [class number](/source/Class_number_(number_theory)) h and the regulator using the [class number formula](/source/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

- The regulator of an [imaginary quadratic field](/source/Imaginary_quadratic_field), or of the rational integers, is 1 (as the determinant of a 0 × 0 matrix is 1).
- The regulator of a [real quadratic field](/source/Real_quadratic_field) is the logarithm of its [fundamental unit](/source/Fundamental_unit_(number_theory)): for example, that of the [golden field](/source/Golden_field) \Q\bigl(\sqrt5~\!\bigr) is \log \tfrac12\bigl(1 + \sqrt{5}~\!\bigr). This can be seen as follows. A fundamental unit is the [golden ratio](/source/Golden_ratio) \tfrac12\bigl(1 + \sqrt{5}~\!\bigr), and its images under the two embeddings into \R are \tfrac12\bigl(1 + \sqrt{5}~\!\bigr) and \tfrac12\bigl(1 - \sqrt{5}~\!\bigr). So the *r* × (*r* + 1) matrix is

\left[1\times\log\left|\frac{\sqrt{5} + 1}{2}\right|, \quad 1\times \log\left|\frac{-\sqrt{5} + 1}{2}\right|\ \right].

- The regulator of the [cyclic cubic field](/source/Cyclic_cubic_field) \mathbb{Q}(\alpha), where α is a root of *x*3 + *x*2 − 2*x* − 1, is approximately 0.5255. A basis of the group of units modulo roots of unity is {*ε*1, *ε*2} where *ε*1 = *α*2 + *α* − 1 and *ε*2 = 2 − *α*2.[4]

## Higher regulators

A 'higher' regulator refers to a construction for a function on an [algebraic K-group](/source/Algebraic_K-group) with index *n* > 1 that plays the same role as the classical regulator does for the group of units, which is a group *K*1. A theory of such regulators has been in development, with work of [Armand Borel](/source/Armand_Borel) and others. Such higher regulators play a role, for example, in the [Beilinson conjectures](/source/Beilinson_conjectures), and are expected to occur in evaluations of certain [L-functions](/source/L-function) at integer values of the argument.[5] See also [Beilinson regulator](/source/Beilinson_regulator).

## Stark regulator

The formulation of [Stark's conjectures](/source/Stark's_conjectures) led [Harold Stark](/source/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](/source/Artin_representation).[6][7]

## p-adic regulator

Let K be a [number field](/source/Number_field) and for each [prime](/source/Valuation_(algebra)) P of K above some fixed rational prime p, let *U**P* denote the local units at P and let *U*1,*P* denote the subgroup of principal units in *U**P*. Set

U_1 = \prod_{P|p} U_{1,P}.

Then let *E*1 denote the set of global units ε that map to *U*1 via the diagonal embedding of the global units in E.

Since *E*1 is a finite-[index](/source/Index_of_a_subgroup) subgroup of the global units, it is an [abelian group](/source/Abelian_group) of rank *r*1 + *r*2 − 1. The **p-adic regulator** is the determinant of the matrix formed by the p-adic logarithms of the generators of this group. *[Leopoldt's conjecture](/source/Leopoldt's_conjecture)* states that this determinant is non-zero.[8][9]

## See also

- [Elliptic unit](/source/Elliptic_unit)
- [Cyclotomic unit](/source/Cyclotomic_unit)
- [Shintani's unit theorem](/source/Shintani's_unit_theorem)

## Notes

1. Elstrodt 2007, §8.D

1. Stevenhagen, P. (2012). [*Number Rings*](http://websites.math.leidenuniv.nl/algebra/ant.pdf). p. 57.

1. Neukirch, Schmidt & Wingberg 2000, proposition VIII.8.6.11.

1. Cohen 1993, Table B.4

1. Bloch, Spencer J. (2000). *Higher regulators, algebraic K-theory, and zeta functions of elliptic curves*. Vol. 11. CRM Monograph Series. Providence, RI: [American Mathematical Society](/source/American_Mathematical_Society). ISBN 0-8218-2114-8. Zbl 0958.19001.

1. Prasad, Dipendra & Yogonanda, C. S. (2007-02-23). [*A Report on Artin's holomorphy conjecture*](http://www.math.tifr.res.in/~dprasad/artin.pdf)

1. Dasgupta, Samit (1999). [*Stark's Conjectures*](https://web.archive.org/web/20080510150747/http://www.math.harvard.edu/~dasgupta/papers/Dasguptaseniorthesis.pdf). Archived from [the original](http://www.math.harvard.edu/~dasgupta/papers/Dasguptaseniorthesis.pdf) on 2008-05-10.

1. Neukirch et al. (2008) p. 626–627

1. Iwasawa, Kenkichi (1972). *Lectures on p-adic L-functions*. Vol. 74. Annals of Mathematics Studies. Princeton, NJ: Princeton University Press and University of Tokyo Press. pp. 36–42. ISBN 0-691-08112-3. Zbl 0236.12001.

## References

- Cohen, Henri (1993). *A Course in Computational Algebraic Number Theory*. Vol. 138. [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics). Berlin, New York: [Springer-Verlag](/source/Springer-Verlag). ISBN 978-3-540-55640-4. MR 1228206. Zbl 0786.11071.
- Elstrodt, Jürgen (2007). ["The Life and Work of Gustav Lejeune Dirichlet (1805–1859)"](https://web.archive.org/web/20210522140235/https://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf). *Clay Mathematics Proceedings*. Archived from [the original](http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf) on 2021-05-22. Retrieved 2010-06-13.
- Lang, Serge (1994). *Algebraic number theory*. Vol. 110. Graduate Texts in Mathematics. 2nd ed. New York: [Springer-Verlag](/source/Springer-Verlag). ISBN 0-387-94225-4. Zbl 0811.11001.

---
Adapted from the Wikipedia article [Dirichlet's unit theorem](https://en.wikipedia.org/wiki/Dirichlet's_unit_theorem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Dirichlet's_unit_theorem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.