Dirichlet's unit theorem

Template:Short description 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 O_K 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 r₁ is the number of real embeddings and r₂ the number of conjugate pairs of complex embeddings of Template:Mvar. This characterisation of r₁ and r₂ is based on the idea that there will be as many ways to embed Template:Mvar in the complex number field as the degree $n=[K:\mathbb{\mathbb{Q}}]$; 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 $\mathbb{\mathbb{Q}}$ then either r₁ = 0 or r₂ = 0.

Other ways of determining r₁ and r₂ are

• use the primitive element theorem to write $K=\mathbb{\mathbb{Q}}(\alpha )$, and then r₁ is the number of conjugates of Template:Mvar that are real, 2r₂ the number that are complex; in other words, if Template:Mvar is the minimal polynomial of Template:Mvar over $\mathbb{\mathbb{Q}}$, then r₁ is the number of real roots and 2r₂ is the number of non-real complex roots of Template:Mvar (which come in complex conjugate pairs);
• write the tensor product of fields $K{\otimes }_{\mathbb{\mathbb{Q}}}\mathbb{ℝ}$ as a product of fields, there being r₁ copies of $\mathbb{ℝ}$ and r₂ copies of $\mathbb{ℂ}$.

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 $\mathbb{\mathbb{Q}}$ 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 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 {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 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 O ⊂ O_K.

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 $\mathbb{\mathbb{Q}}\oplus O_{K,S}{\otimes }_{\mathbb{\mathbb{Z}}}\mathbb{\mathbb{Q}}$ has been determined.Template:Sfn

The regulator

Suppose that K is a number field and $u_1,\dots ,u_r$ are a set of generators for the unit group of K modulo roots of unity. There will be r + 1 Archimedean places of K, either real or complex. For $u\in K$, write $u^{(1)},\dots ,u^{(r+1)}$ for the different embeddings into $\mathbb{ℝ}$ or $\mathbb{ℂ}$ and set N_j to 1 or 2 if the corresponding embedding is real or complex respectively. Then the r × (r + 1) matrix $${(N_j\log \left|u_i^{(j)}\right|)}_{i=1,\dots ,r,j=1,\dots ,r+1}$$ 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 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 Template:Mvar to the vector with entries $N_j\log \left|u^{(j)}\right|$ has an image in the Template:Mvar-dimensional subspace of ${\mathbb{ℝ}}^{r+1}$ 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 $R\sqrt{r+1}$.

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 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.

Examples

File:Discriminant49CubicFieldFundamentalDomainOfUnits.png

A fundamental domain in logarithmic space of the group of units of the cyclic cubic field Template:Mvar obtained by adjoining to $\mathbb{\mathbb{Q}}$ a root of f(x) = x³ + x² − 2x − 1. If Template:Mvar 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 Template:Mvar 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 a real quadratic field is the logarithm of its fundamental unit: for example, that of the golden field $\mathbb{\mathbb{Q}}\left(\sqrt{5}\right)$ is $\log \frac{1}{2}(1+\sqrt{5})$. This can be seen as follows. A fundamental unit is the golden ratio $\frac{1}{2}(1+\sqrt{5})$, and its images under the two embeddings into $\mathbb{ℝ}$ are $\frac{1}{2}(1+\sqrt{5})$ and $\frac{1}{2}(1-\sqrt{5})$. So the r × (r + 1) matrix is $$[1\times \log \left|\frac{\sqrt{5}+1}{2}\right|,1\times \log \left|\frac{-\sqrt{5}+1}{2}\right|].$$
• The regulator of the cyclic cubic field $\mathbb{\mathbb{Q}}(\alpha )$, where Template:Mvar 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 a function on an [[algebraic K-group|algebraic Template:Mvar-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-function|Template:Mvar-function]]s 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.

Template:Mvar-adic regulator

Let Template:Mvar be a number field and for each prime Template:Mvar of Template:Mvar above some fixed rational prime Template:Mvar, let U_P denote the local units at Template:Mvar and let U_1,P denote the subgroup of principal units in U_P. Set $$U_1=\prod _{P|p}{U}_{1,P}.$$

Then let E₁ denote the set of global units Template:Mvar that map to U₁ via the diagonal embedding of the global units in Template:Mvar.

Since E₁ is a finite-index subgroup of the global units, it is an abelian group of rank r₁ + r₂ − 1. 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.

See also

• Elliptic unit
• Cyclotomic unit
• Shintani's unit theorem

References

• Template:Neukirch et al. CNF