In number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function

Dedekind zeta function

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK, is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q...

General statement of the class number formula

Let K be a number field with , where denotes the number of real embeddings of K, and is the number of complex embeddings of Let


be the Dedekind zeta function of Also define the following invariants

Invariant (mathematics)

In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

:

• is the class number, the number of elements in the ideal class group of
• is the regulator of
• is the number of roots of unity
  Root of unity
  In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
  
   contained in
• is the discriminant
  Discriminant of an algebraic number field
  In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the algebraic number field...
  
   of the extension
  Algebraic extension
  In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
  
   

Then:

Theorem 1 (Class Number Formula) The Dedekind zeta function

Dedekind zeta function

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK, is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q...

 of K, converges absolutely for and extends to a meromorphic function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 defined for all complex s with only one simple pole at with residue


This is the most general "class number formula". In particular cases, for example when K is a cyclotomic extension of , there are particular and more refined class number formulas.

Galois extensions of the rationals

If K is a Galois extension

Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...

 of Q, the theory of Artin L-function

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 the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...

s applies to . It has one factor of the Riemann zeta function, 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 complex linear representations of Gal(K/Q) of dimension dim(ρ). That is according to the standard decomposition of the regular representation

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....

.

Abelian extensions of the rationals

This is the case of the above, with Gal(K/Q) an abelian group

Abelian group

In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...

, in which all the ρ can be replaced by Dirichlet character

Dirichlet character

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...

s (via class field theory

Class field theory

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...

) for some modulus f called the conductor. Therefore all the L(1) values occur for Dirichlet L-functions, for which there is a classical formula, involving logarithms.

By the Kronecker–Weber theorem

Kronecker–Weber theorem

In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...

, 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

Ernst Kummer

Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...

. 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 unit

Cyclotomic unit

In mathematics, a cyclotomic unit is a unit of an algebraic number field of the form / for ζ a root of unity, or more generally a unit that can be written as a product of these and a root of unity....

s. 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

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 Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur...

, these ideas are further combined with Stickelberger's theorem

Stickelberger's theorem

In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields...

.

Dirichlet class number formula

This exposition follows Davenport.
The first class number formula was proved by Dirichlet in 1839, but it was proved about classes of quadratic forms rather than classes of ideal

Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

s. Let d be a fundamental discriminant

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. Conversely, every integer D with is the discriminant of some binary quadratic form with integer...

, and write for the number of equivalence classes of quadratic forms with discriminant d. Let be the Kronecker symbol

Kronecker symbol

In number theory, the Kronecker symbol, written as \left or , is a generalization of the Jacobi symbol to all integers n. It was introduced by Leopold Kronecker.-Definition:...

. Then is a Dirichlet character

Dirichlet character

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...

. Write for the Dirichlet L-series based on . For , let , be the solution to the Pell equation  for which u is smallest, and write
(Then is either a fundamental unit

Fundamental unit (number theory)

In algebraic number theory, a fundamental unit is a generator for the torsion-free unit group of the ring of integers of a number field, when that group is infinite cyclic...

 of the real quadratic field  or the square of a fundamental unit.)
For , write w for the number of automorphs 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

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...

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. We have