Dirichlet's unit theorem
Encyclopedia
In mathematics
, Dirichlet's unit theorem is a basic result in algebraic number theory
due to Gustav Lejeune Dirichlet. It determines the rank
of the group of units in the ring
OK of algebraic integer
s of a number field K. The regulator is a positive real number that determines how "dense" the units are.
(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 n = [K : Q]; these will either be into the real number
s, or pairs of embeddings related by complex conjugation, so that
Other ways of determining r1 and r2 are
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 > 0 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.
The torsion in the group of units is the set of all roots of unity of K, which form a finite 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, 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.)
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
s, determining the rank of the unit group in localizations
of rings of integers. Also, the Galois module
structure of
has been determined.
Ni for the degree of the corresponding embedding over R (so it is 1 for real embeddings and 2 for complex ones).
Then the r by r + 1 matrix whose entries are
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 of 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 ui). 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 Nilog|ui| has image in the r-dimensional subspace of Rr+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√(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 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.
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
s at integer values of the argument.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Dirichlet's unit theorem is a basic result in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
due to Gustav Lejeune Dirichlet. It determines the rank
Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
of the group of units in the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
OK of algebraic integer
Algebraic integer
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...
s of a number field K. The regulator is a positive real number that determines how "dense" the units are.
Dirichlet's unit theorem
The statement is that the group of units is finitely generated and has rankRank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
(maximal number of multiplicatively independent elements) equal to
- r = r1 + r2 − 1
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
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
field as the degree n = [K : Q]; these will either be into the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, or pairs of embeddings related by complex conjugation, so that
- n = r1 + 2r2.
Other ways of determining r1 and r2 are
- use the primitive element theorem to write K = Q(α), and then r1 is the number of conjugatesConjugate element (field theory)In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the roots of the minimal polynomialof α over K.-Example:The cube roots of the number one are:...
of α that are real, 2r2 the number that are complex;
- write the tensor product of fieldsTensor product of fieldsIn abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is never itself a field. On the other hand it is often required to 'join' two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K...
K ⊗QR as a product of fields, there being r1 copies of R and r2 copies of C.
As an example, if K is a quadratic field
Quadratic field
In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields...
, 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
Pell's equation
Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...
.
The rank is > 0 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
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
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
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
. 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, 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.)
There is a generalisation of the unit theorem by Helmut Hasse
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
(and later Claude Chevalley
Claude Chevalley
Claude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups...
) to describe the structure of the group of S-unit
S-unit
In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for S-units.-Definition:...
s, determining the rank of the unit group in localizations
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
of rings of integers. Also, the Galois module
Galois module
In mathematics, a Galois module is a G-module where G is the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module...
structure of
has been determined.The regulator
Suppose that u1,...,ur are a set of generators for the unit group modulo roots of unity. If u is an algebraic number, write u1, ..., ur+1 for the different embeddings into R or C, and writeNi for the degree of the corresponding embedding over R (so it is 1 for real embeddings and 2 for complex ones).
Then the r by r + 1 matrix whose entries are
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 of 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 ui). 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 Nilog|ui| has image in the r-dimensional subspace of Rr+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√(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 h and the regulator using the class number formula
Class number formula
In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function-General statement of 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.
Examples
- The regulator of an imaginary quadratic field, or of the rational integers, is 1 (as the determinant of a 0×0 matrix is 1).
- The regulator of the real quadratic field Q(√5) is log((√5 + 1)/2). This can be seen as follows. A fundamental unit is (√5 + 1)/2, and its images under the two embeddings into R are (√5 + 1)/2 and (−√5 + 1)/2. So the r by r + 1 matrix is
- The regulator of the cyclic cubic field Q(α), where α is a root of x3 + x2 − 2x − 1, is approximately 0.5255. A basis of the group of units modulo roots of unity is {ε1, ε2} where ε1 = α2 + α − 1 and ε2 = 2 − α2.
Higher regulators
A 'higher' regulator refers to a construction for 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 BorelArmand Borel
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...
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
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s at integer values of the argument.
Stark regulator
The formulation of Stark's conjectures led Harold StarkHarold Stark
Harold Mead Stark is an American mathematician, specializing in number theory. He is best known for his solution of the Gauss class number 1 problem, in effect correcting and completing the earlier work of Kurt Heegner; and for Stark's conjecture. He has recently collaborated with Audrey Terras on...
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.