Dirichlet's unit theorem
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . 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 is the "number of real embeddings" and the "number of conjugate pairs of complex embeddings" of . This characterisation of and  is based on the idea that there will be as many ways to embed in the complex number field as the degree formula_1; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that Note that if is Galois over formula_2 then either or . Other ways of determining and are As an example, if 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 formula_2 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 is large. The torsion in the group of units is the set of all roots of unity of , which form a finite cyclic group. For a number field with at least one real embedding the torsion must therefore be only . There are number fields, for example most imaginary quadratic fields, having  no real embeddings which also have for the torsion of its unit group. Totally real fields are special with respect to units. If is a finite extension of number fields with degree greater than 1 and the units groups for the integers of and have the same rank then is totally real and is a totally complex quadratic extension. The converse holds too. (An example is equal to the rationals and equal to an imaginary quadratic field; both have unit rank 0.) The theorem not only applies to the maximal order but to any order . There is a generalisation of  the unit theorem by Helmut Hasse (and later Claude Chevalley) to describe the structure of the group of "-units", determining the rank of the unit group in localizations of rings of integers. Also, the Galois module structure of formula_9 has been determined. The regulator. Suppose that "K" is a number field and formula_10 are a set of generators for the unit group of "K" modulo roots of unity. There will be Archimedean places of "K", either real or complex. For formula_11, write formula_12 for the different embeddings into formula_6 or formula_7 and set to 1 or 2 if the corresponding  embedding is real or complex respectively. Then the matrixformula_15has 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 of the determinant of the submatrix formed by deleting one column is independent of the column. The number is called the regulator of the algebraic number field (it does not depend on the choice of generators ). 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 to the vector with entries formula_16 has an image in the -dimensional subspace of formula_17 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 formula_18. 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 	Dirichlet's unit theorem
25931439	to calculate the product of the class number 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. Higher regulators. A 'higher' regulator refers to a construction for a function on an algebraic -group with index that plays the same role as the classical regulator does for the group of units, which is a group . 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 	Dirichlet's unit theorem
25931440	Beilinson conjectures, and are expected to occur in evaluations of certain -functions at integer values of the argument. 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. Section::::-adic regulator. Let be a number field and for each prime of above some fixed rational prime , let denote the local units at and let denote the subgroup of principal units in . Set formula_27 Then let denote the set of global 	Dirichlet's unit theorem
25931441	units that map to via the diagonal embedding of the global units in . Since is a finite-index subgroup of the global units, it is an abelian group of rank . The -adic regulator is the determinant of the matrix formed by the -adic logarithms of the generators of this group. Leopoldt's conjecture" states that this determinant is non-zero.