Class number formula

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

Let $K$ be a number field with $[K:\mathbb{Q}]=n=r_1+2r_2$ , where $r 1$ denotes the number of real embeddings of $K$ , and $2 r 2$ is the number of complex embeddings of $K .$ Let

$\zeta_K(s) \,$

be the Dedekind zeta function of $K .$ Also define the following invariants:

$h K$ is the class number, the number of elements in the ideal class group of $K .$
$\operatorname{Reg}_K$ is the regulator of $K .$
$w K$ is the number of roots of unity contained in $K .$
$D K$ is the discriminant of the extension $K/\mathbb{Q}.$

Then:

Theorem 1 (Class Number Formula) The Dedekind zeta function of $K$ , $ζ K (s)$ converges absolutely for $\Re(s)>1$ and extends to a meromorphic function defined for all complex s with only one simple pole at $s = 1,$ with residue

$\lim_{s\to 1} (s-1)\zeta_K(s)=\frac{2^{r_1}\cdot(2\pi)^{r_2}\cdot h_K\cdot \operatorname{Reg}_K}{w_K \cdot \sqrt{\mid D_K \mid}}$

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

[edit] Galois extensions of the rationals

If K is a Galois extension of Q, the theory of Artin L-functions applies to $ζ K (s)$ . 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.

[edit] Abelian extensions of the rationals

This is the case of the above, with Gal(K/Q) an abelian group, in which all the ρ can be replaced by Dirichlet characters (via class field theory) 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, 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. 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, these ideas are further combined with Stickelberger's theorem.

This article incorporates material from Class number formula on PlanetMath, which is licensed under the GFDL.

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