Template:Short description In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet.<ref>Template:Harvnb</ref> It determines the rank of the group of units in the ring Template:Math of algebraic integers of a number field Template:Mvar. 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 Template:Block indent where Template:Math is the number of real embeddings and Template:Math the number of conjugate pairs of complex embeddings of Template:Mvar. This characterisation of Template:Math and Template:Math is based on the idea that there will be as many ways to embed Template:Mvar in the complex number field as the degree <math>n = [K: \mathbb{Q}]</math>; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that Template:Block indent
Note that if Template:Mvar is Galois over <math>\mathbb{Q}</math> then either Template:Math or Template:Math.
Other ways of determining Template:Math and Template:Math are
- use the primitive element theorem to write <math>K = \mathbb{Q}(\alpha)</math>, and then Template:Math is the number of conjugates of Template:Mvar that are real, Template:Math the number that are complex; in other words, if Template:Mvar is the minimal polynomial of Template:Mvar over <math>\mathbb{Q}</math>, then Template:Math is the number of real roots and Template:Math is the number of non-real complex roots of Template:Mvar (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 Template:Math copies of <math>\mathbb{R}</math> and Template:Math copies of <math>\mathbb{C}</math>.
As an example, if Template:Mvar 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 Template:Mvar is large.
The torsion in the group of units is the set of all roots of unity of Template:Mvar, which form a finite cyclic group. For a number field with at least one real embedding the torsion must therefore be only Template:Math. There are number fields, for example most imaginary quadratic fields, having no real embeddings which also have Template:Math for the torsion of its unit group.
Totally real fields are special with respect to units. If Template:Math is a finite extension of number fields with degree greater than 1 and the units groups for the integers of Template:Mvar and Template:Mvar have the same rank then Template:Mvar is totally real and Template:Mvar is a totally complex quadratic extension. The converse holds too. (An example is Template:Mvar equal to the rationals and Template:Mvar equal to an imaginary quadratic field; both have unit rank 0.)
The theorem not only applies to the maximal order Template:Mvar but to any order Template:Math.<ref>Template:Cite book</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|Template:Mvar-unit]]s, determining the rank of the unit group in 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.Template:Sfn
The regulatorEdit
Suppose that K is a number field and <math>u_1, \dots, u_r</math> are a set of generators for the unit group of K modulo roots of unity. There will be Template:Math Archimedean places of K, either real or complex. For <math>u\in K</math>, write <math>u^{(1)},\dots,u^{(r+1)}</math> for the different embeddings into <math>\mathbb{R}</math> or <math>\mathbb{C}</math> and set Template:Math to 1 or 2 if the corresponding embedding is real or complex respectively. Then the Template:Math matrix <math display="block">\left(N_j\log \left|u_i^{(j)}\right|\right)_{i=1,\dots,r,\; j=1,\dots,r+1}</math> 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 Template:Mvar of the determinant of the submatrix formed by deleting one column is independent of the column. The number Template:Mvar is called the regulator of the algebraic number field (it does not depend on the choice of generators Template:Math). 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 Template:Mvar to the vector with entries <math display="inline">N_j\log \left|u^{(j)}\right|</math> has an image in the Template:Mvar-dimensional subspace of <math>\mathbb{R}^{r + 1}</math> 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 <math>R\sqrt{r + 1}</math>.
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 Template:Math of the class number Template:Mvar 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.
ExamplesEdit
- The regulator of an imaginary quadratic field, or of the rational integers, is 1 (as the determinant of a Template:Math matrix is 1).
- The regulator of a real quadratic field is the logarithm of its fundamental unit: for example, that of <math>\mathbb{Q}(\sqrt{5})</math> is <math display="inline">\log \frac{\sqrt{5} + 1}{2}</math>. This can be seen as follows. A fundamental unit is <math display="inline">(\sqrt{5} + 1) / 2</math>, and its images under the two embeddings into <math>\mathbb{R}</math> are <math display="inline">(\sqrt{5} + 1) / 2</math> and <math display="inline">(-\sqrt{5} + 1) / 2</math>. So the Template:Math matrix is <math display="block">\left[1\times\log\left|\frac{\sqrt{5} + 1}{2}\right|, \quad 1\times \log\left|\frac{-\sqrt{5} + 1}{2}\right|\ \right].</math>
- The regulator of the cyclic cubic field <math>\mathbb{Q}(\alpha)</math>, where Template:Mvar is a root of Template:Math, is approximately 0.5255. A basis of the group of units modulo roots of unity is Template:Math where Template:Math and Template:Math.<ref>Template:Harvnb</ref>
Higher regulatorsEdit
A 'higher' regulator refers to a construction for a function on an [[algebraic K-group|algebraic Template:Mvar-group]] with index Template:Math that plays the same role as the classical regulator does for the group of units, which is a group Template:Math. 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-function|Template:Mvar-function]]s at integer values of the argument.<ref name=Bloch>Template:Cite book</ref> See also Beilinson regulator.
Stark regulatorEdit
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.<ref>Template:Cite report</ref><ref>Template:Cite thesis</ref>
Template:Mvar-adic regulatorEdit
Let Template:Mvar be a number field and for each prime Template:Mvar of Template:Mvar above some fixed rational prime Template:Mvar, let Template:Math denote the local units at Template:Mvar and let Template:Math denote the subgroup of principal units in Template:Math. Set <math display="block"> U_1 = \prod_{P|p} U_{1,P}. </math>
Then let Template:Math denote the set of global units Template:Mvar that map to Template:Math via the diagonal embedding of the global units in Template:Mvar.
Since Template:Math is a finite-index subgroup of the global units, it is an abelian group of rank Template:Math. The Template:Mvar-adic regulator is the determinant of the matrix formed by the Template:Mvar-adic logarithms of the generators of this group. Leopoldt's conjecture states that this determinant is non-zero.<ref name=NSW6267>Neukirch et al. (2008) p. 626–627</ref><ref>Template:Cite book</ref>