# Dirichlet's unit theorem

In mathematics, **Dirichlet's unit theorem** is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet.^{[1]} It determines the rank of the group of units in the ring *O*_{K} of algebraic integers of a 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 (maximal number of multiplicatively independent elements) equal to

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 field as the degree *n* = [*K* : ℚ]; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that

As an example, if 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 ℚ 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 n is large.

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 O_{K} but to any order *O* ⊂ *O _{K}*

*.*

^{[2]}There is a generalisation of the unit theorem by Helmut Hasse (and later Claude Chevalley) to describe the structure of the group of *S-units*, determining the rank of the unit group in localizations of rings of integers. Also, the Galois module structure of ℚ ⊕ *O*_{K,S} ⊗_{ℤ} ℚ has been determined.^{[3]}

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 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.

A 'higher' regulator refers to a construction for a function on an 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 and others. Such higher regulators play a role, for example, in the Beilinson conjectures, and are expected to occur in evaluations of certain L-functions at integer values of the argument.^{[5]} See also Beilinson 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.^{[6]}^{[7]}

Let K be a number field and for each prime 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

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 subgroup of the global units, it is an 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* states that this determinant is non-zero.^{[8]}^{[9]}