In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
, the class number formula relates many important invariants of a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
to a special value of its
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
.
General statement of the class number formula
We start with the following data:
* is a number field.
* , where denotes the number of
real embeddings of , and is the number of complex embeddings of .
* is the
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
of .
* is the
class number, the number of elements in the
ideal class group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
of .
* is the
regulator of .
* is the number of
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
contained in .
* is the
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
of the
extension .
Then:
:Theorem (Class Number Formula).
converges absolutely for and extends to a
meromorphic
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pol ...
function defined for all complex with only one
simple pole at , with residue
::
This is the most general "class number formula". In particular cases, for example when is a
cyclotomic extension of , there are particular and more refined class number formulas.
Proof
The idea of the proof of the class number formula is most easily seen when ''K'' = Q(i). In this case, the ring of integers in ''K'' is the
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s.
An elementary manipulation shows that the residue of the Dedekind zeta function at ''s'' = 1 is the average of the coefficients of the
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in ana ...
representation of the Dedekind zeta function. The ''n''-th coefficient of the Dirichlet series is essentially the number of representations of ''n'' as a sum of two squares of nonnegative integers. So one can compute the residue of the Dedekind zeta function at ''s'' = 1 by computing the average number of representations. As in the article on the
Gauss circle problem
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem is ...
, one can compute this by approximating the number of lattice points inside of a quarter circle centered at the origin, concluding that the residue is one quarter of pi.
The proof when ''K'' is an arbitrary imaginary quadratic number field is very similar.
In the general case, by
Dirichlet's unit theorem
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 of algebraic integers of a number field . The regulator is a pos ...
, the group of units in the ring of integers of ''K'' is infinite. One can nevertheless reduce the computation of the residue to a lattice point counting problem using the classical theory of real and complex embeddings and approximate the number of lattice points in a region by the volume of the region, to complete the proof.
Dirichlet class number formula
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
published a proof of the class number formula for
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
s in 1839, but it was stated in the language of
quadratic forms
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
rather than classes of
ideals. It appears that Gauss already knew this formula in 1801.
This exposition follows
Davenport.
[
]
Let ''d'' be a
fundamental discriminant In mathematics, a fundamental discriminant ''D'' is an integer invariant in the theory of integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of ''Q''(''x'', ''y''). Conversely, every intege ...
, and write ''h(d)'' for the number of equivalence classes of quadratic forms with discriminant ''d''. Let
be the
Kronecker symbol
In number theory, the Kronecker symbol, written as \left(\frac an\right) or (a, n), is a generalization of the Jacobi symbol to all integers n. It was introduced by .
Definition
Let n be a non-zero integer, with prime factorization
:n=u \cdot ...
. Then
is a
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \ch ...
. Write
for the
Dirichlet L-series
In mathematics, a Dirichlet ''L''-series is a function of the form
:L(s,\chi) = \sum_^\infty \frac.
where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By ...
based on
. For ''d > 0'', let ''t > 0'', ''u > 0'' be the solution to the
Pell equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates ...
for which ''u'' is smallest, and write
:
(Then
is either a
fundamental unit of the
real quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
or the square of a fundamental unit.)
For ''d'' < 0, write ''w'' for the number of automorphisms of quadratic forms of discriminant ''d''; that is,
:
Then Dirichlet showed that
:
This is a special case of Theorem 1 above: for a
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
''K'', the Dedekind zeta function is just
, and the residue is
. Dirichlet also showed that the ''L''-series can be written in a finite form, which gives a finite form for the class number. Suppose
is
primitive with prime
conductor . Then
:
Galois extensions of the rationals
If ''K'' is a
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
of Q, the theory of
Artin L-function In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. T ...
s applies to
. It has one factor of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, which has a pole of residue one, and the quotient is regular at ''s'' = 1. This means that the right-hand side of the class number formula can be equated to a left-hand side
:Π ''L''(1,ρ)
dim ρ
with ρ running over the classes of irreducible non-trivial complex
linear representation
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
s of Gal(''K''/Q) of dimension dim(ρ). That is according to the standard decomposition of the
regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation.
One distinguishes the left regular re ...
.
Abelian extensions of the rationals
This is the case of the above, with Gal(''K''/Q) an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
, in which all the ρ can be replaced by
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \ch ...
s (via
class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
) for some modulus ''f'' called the
conductor. Therefore all the ''L''(1) values occur for
Dirichlet L-function
In mathematics, a Dirichlet ''L''-series is a function of the form
:L(s,\chi) = \sum_^\infty \frac.
where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. B ...
s, for which there is a classical formula, involving logarithms.
By the
Kronecker–Weber theorem
In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provides a partial conver ...
, all the values required for an analytic class number formula occur already when the cyclotomic fields are considered. In that case there is a further formulation possible, as shown by
Kummer Kummer is a German surname. Notable people with the surname include:
* Bernhard Kummer (1897–1962), German Germanist
* Clare Kummer (1873—1958), American composer, lyricist and playwright
* Clarence Kummer (1899–1930), American jockey
* Chri ...
. The
regulator, a calculation of volume in 'logarithmic space' as divided by the logarithms of the units of the cyclotomic field, can be set against the quantities from the ''L''(1) recognisable as logarithms of
cyclotomic units. There result formulae stating that the class number is determined by the index of the cyclotomic units in the whole group of units.
In
Iwasawa theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the ea ...
, these ideas are further combined with
Stickelberger's theorem.
Notes
References
*
{{L-functions-footer
Algebraic number theory
Quadratic forms