Dirichlet's unit theorem

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

The statement is that the group of units is finitely generated and has rank (maximal number of multiplicatively independent elements) equal to

r = r 1 + r 2 - 1

where $r 1$ is the number of real embeddings and $r 2$ the number of conjugate pairs of complex embeddings of $K$ . This characterisation of $r 1$ and $r 2$ is based on the idea that there will be as many ways to embed $K$ in the complex number field as the degree $n=[K:\mathbb {Q} ]$ ; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that

n = r 1 + 2 r 2

.

Note that if $K$ is Galois over $\mathbb {Q}$ then either $r 1 = 0$ or $r 2 = 0$ .

Other ways of determining $r 1$ and $r 2$ are

use the primitive element theorem to write $K=\mathbb {Q} (\alpha )$ , and then $r 1$ is the number of conjugates of $α$ that are real, $2 r 2$ the number that are complex; in other words, if $f$ is the minimal polynomial of $α$ over $\mathbb {Q}$ , then $r 1$ is the number of real roots and $2r 2$ is the number of non-real complex roots of $f$ (which come in complex conjugate pairs);
write the tensor product of fields $K\otimes _{\mathbb {Q} }\mathbb {R}$ as a product of fields, there being $r 1$ copies of $\mathbb {R}$ and $r 2$ copies of $\mathbb {C}$ .

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 positive for all number fields besides $\mathbb {Q}$ 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 $n$ 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.)

The theorem not only applies to the maximal order $O K$ but to any order $O \subset O K$ .^[2]

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 $\mathbb {Q} \oplus O_{K,S}\otimes _{\mathbb {Z} }\mathbb {Q}$ has been determined.^[3]

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 $r \times (r + 1)$ matrix $\left(N_{j}\log \left|u_{i}^{(j)}\right|\right)_{i=1,\dots ,r,\;j=1,\dots ,r+1}$ 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 $R$ 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 ${\textstyle N_{j}\log \left|u^{(j)}\right|}$ has an image in the $r$ -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 $h$ 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 $\mathbb {Q}$ a root of $f (x) = x 3 + x 2 - 2 x - 1$ . If $α$ denotes a root of $f (x)$ , then a set of fundamental units is ${ε 1, ε 2}$ , where $ε 1 = α 2 + α - 1$ and $ε 2 = 2 - α 2$ . 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 $0 \times 0$ matrix is 1).
The regulator of a real quadratic field is the logarithm of its fundamental unit: for example, that of $\mathbb {Q} ({\sqrt {5}})$ is ${\textstyle \log {\frac {{\sqrt {5}}+1}{2}}}$ . This can be seen as follows. A fundamental unit is ${\textstyle ({\sqrt {5}}+1)/2}$ , and its images under the two embeddings into $\mathbb {R}$ are ${\textstyle ({\sqrt {5}}+1)/2}$ and ${\textstyle (-{\sqrt {5}}+1)/2}$ . So the $r \times (r + 1)$ matrix is $\left[1\times \log \left|{\frac {{\sqrt {5}}+1}{2}}\right|,\quad 1\times \log \left|{\frac {-{\sqrt {5}}+1}{2}}\right|\ \right].$
The regulator of the cyclic cubic field $\mathbb {Q} (\alpha )$ , where $α$ is a root of $x 3 + x 2 - 2 x - 1$ , is approximately 0.5255. A basis of the group of units modulo roots of unity is ${ε 1, ε 2}$ where $ε 1 = α 2 + α - 1$ and $ε 2 = 2 - α 2$ .^[4]

Higher regulators

A 'higher' regulator refers to a construction for a function on 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 $K 1$ . 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.^[5] 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.^[6]^[7]

$p$ -adic regulator

Let $K$ be a number field and for each prime $P$ of $K$ above some fixed rational prime $p$ , let $U P$ denote the local units at $P$ 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 $E 1$ denote the set of global units $ε$ that map to $U 1$ via the diagonal embedding of the global units in $E$ .

Since $E 1$ is a finite-index subgroup of the global units, it is an abelian group of rank $r 1 + r 2 - 1$ . The $p$ -adic regulator is the determinant of the matrix formed by the $p$ -adic logarithms of the generators of this group. Leopoldt's conjecture states that this determinant is non-zero.^[8]^[9]

Notes

^ Elstrodt 2007, §8.D
^ Stevenhagen, P. (2012). Number Rings (PDF). p. 57.
^ Neukirch, Schmidt & Wingberg 2000, proposition VIII.8.6.11.
^ Cohen 1993, Table B.4
^ Bloch, Spencer J. (2000). Higher regulators, algebraic $K$ -theory, and zeta functions of elliptic curves. CRM Monograph Series. Vol. 11. Providence, RI: American Mathematical Society. ISBN 0-8218-2114-8. Zbl 0958.19001.
^ Prasad, Dipendra; Yogonanda, C. S. (2007-02-23). A Report on Artin's holomorphy conjecture (PDF) (Report).
^ Dasgupta, Samit (1999). Stark's Conjectures (PDF) (Thesis). Archived from the original (PDF) on 2008-05-10.
^ Neukirch et al. (2008) p. 626–627
^ Iwasawa, Kenkichi (1972). Lectures on $p$ -adic $L$ -functions. Annals of Mathematics Studies. Vol. 74. Princeton, NJ: Princeton University Press and University of Tokyo Press. pp. 36–42. ISBN 0-691-08112-3. Zbl 0236.12001.

References

Cohen, Henri (1993). A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics. Vol. 138. Berlin, New York: Springer-Verlag. ISBN 978-3-540-55640-4. MR 1228206. Zbl 0786.11071.
Elstrodt, Jürgen (2007). "The Life and Work of Gustav Lejeune Dirichlet (1805–1859)" (PDF). Clay Mathematics Proceedings. Archived from the original (PDF) on 2021-05-22. Retrieved 2010-06-13.
Lang, Serge (1994). Algebraic number theory. Graduate Texts in Mathematics. Vol. 110 (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94225-4. Zbl 0811.11001.
Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001

[1] Elstrodt 2007, §8.D

[2] Stevenhagen, P. (2012). Number Rings (PDF). p. 57.

[FOOTNOTENeukirchSchmidtWingberg2000proposition_VIII.8.6.11-3] Neukirch, Schmidt & Wingberg 2000, proposition VIII.8.6.11.

[4] Cohen 1993, Table B.4

[Bloch-5] Bloch, Spencer J. (2000). Higher regulators, algebraic $K$ -theory, and zeta functions of elliptic curves. CRM Monograph Series. Vol. 11. Providence, RI: American Mathematical Society. ISBN 0-8218-2114-8. Zbl 0958.19001.

[6] Prasad, Dipendra; Yogonanda, C. S. (2007-02-23). A Report on Artin's holomorphy conjecture (PDF) (Report).

[7] Dasgupta, Samit (1999). Stark's Conjectures (PDF) (Thesis). Archived from the original (PDF) on 2008-05-10.

[NSW6267-8] Neukirch et al. (2008) p. 626–627

[9] Iwasawa, Kenkichi (1972). Lectures on $p$ -adic $L$ -functions. Annals of Mathematics Studies. Vol. 74. Princeton, NJ: Princeton University Press and University of Tokyo Press. pp. 36–42. ISBN 0-691-08112-3. Zbl 0236.12001.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]