Dirichlet's unit theorem

In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to 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.

Dirichlet's unit theorem

The statement is that the group of units is finitely generated and has rank (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 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

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 conjugates of α that are real, 2r₂ the number that are complex;

• write the tensor product of fields 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, 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-units, determining the rank of the unit group in localizations of rings of integers. Also, the Galois module structure of $\mathbf{Q} \oplus O_{K,S} \otimes_\mathbf{Z} \mathbf{Q}$ has been determined.^[2]

The regulator

Suppose that u₁,...,u_r are a set of generators for the unit group modulo roots of unity. If u is an algebraic number, write u¹, ..., u^r+1 for the different embeddings into R or C, and write N_i 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 $N_i\log|u_i^j|$ 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 u_i). 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 N_ilog|u_i| has image in the r-dimensional subspace of R^r+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.

Examples

A fundamental domain in logarithmic space of the group of units of the cyclic cubic field K obtained by adjoining to Q a root of f(x) = x³ + x² − 2x − 1. If α denotes a root of f(x), then a set of fundamental units is {ε₁, ε₂} where ε₁ = α² + α − 1 and ε₂ = 2 − α². The area of the fundamental domain is approximately 0.910114, so the regulator of K is approximately 0.525455.

• 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

$\left[1\times\log\left|{\sqrt{5} + 1 \over 2}\right|, \quad 1\times \log\left|{-\sqrt{5} + 1 \over 2}\right|\ \right].$

• 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.^[3]

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 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.

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.^[4][5]

See also

• Elliptic unit
• Cyclotomic unit

Notes

1. ^ Elstrodt 2007, §8.D
2. ^ Neukirch, Schmidt & Wingberg 2000, proposition VIII.8.6.11.
3. ^ Cohen 1993, Table B.4
4. ^ PDF
5. ^ PDF

References

• Cohen, Henri (1993). A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics. 138. Berlin, New York: Springer-Verlag. ISBN 978-3-540-55640-4. MR1228206 
• Elstrodt, Jürgen (2007). "The Life and Work of Gustav Lejeune Dirichlet (1805–1859)" (PDF). Clay Mathematics Proceedings. http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf. Retrieved 2010-06-13. 
• Serge Lang, Algebraic number theory, ISBN 0-387-94225-4
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR1697859 
• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR1737196