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 OK 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 r1 is the number of real embeddings and r2 the number of conjugate pairs of complex embeddings of K. This characterisation of r1 and r2 is based on the idea that there will be as many ways to embed K in the complex number field as the degree

; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that Note that if K is Galois over

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.

For a number field with at least one real embedding the torsion must therefore be only {1,−1}.

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 theorem not only applies to the maximal order OK but to any order O ⊂ OK.[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.

are a set of generators for the unit group of K modulo roots of unity.

and set Nj to 1 or 2 if the corresponding embedding is real or complex respectively.

The map taking a unit u to the vector with entries

consisting of all vectors whose entries have sum 0, and by Dirichlet's unit theorem the image is a lattice in this subspace.

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.

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 K1.

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.

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 UP denote the local units at P and let U1,P denote the subgroup of principal units in UP.

Then let E1 denote the set of global units ε that map to U1 via the diagonal embedding of the global units in E. Since E1 is a finite-index subgroup of the global units, it is an abelian group of rank r1 + r2 − 1.

Leopoldt's conjecture states that this determinant is non-zero.

A fundamental domain in logarithmic space of the group of units of the cyclic cubic field K obtained by adjoining to a root of f ( x ) = x 3 + x 2 − 2 x − 1 . If α denotes a root of f ( x ) , then a set of fundamental units is { ε 1 , ε 2 } , where ε 1 = α 2 + α − 1 and ε 2 = 2 − α 2 . The area of the fundamental domain is approximately 0.910114, so the regulator of K is approximately 0.525455.