Class number formula

In the mathematical subfield of algebraic number theory , the class number formula gives a formula for calculating the class number of a number field . It was proved for square number fields in 1839 by Peter Gustav Lejeune Dirichlet .

Basics

A number field is a finite field extension of the field of rational numbers . The wholeness ring consists of those elements that can be obtained as the solution of a normalized polynomial equation with integer coefficients. The ideal class group measures how far the wholeness ring is from having a unique prime factorization. It is defined as the group of broken ideals modulo of the broken main ideals. The class number of the number field is defined as the number of elements of the ideal class group. In particular, the class number is if and only if a main ideal ring is and this in turn is the case if and only if the prime factorization is unique. A central problem of algebraic number theory is the question of which number fields have class numbers . ${\displaystyle K}$ ${\displaystyle K/\mathbb {Q} }$ ${\displaystyle {\mathcal {O}}_{K}\subset K}$ ${\displaystyle K}$ ${\displaystyle h_{K}}$ ${\displaystyle 1}$ ${\displaystyle {\mathcal {O}}_{K}}$ ${\displaystyle {\mathcal {O}}_{K}}$ ${\displaystyle h_{K}=1}$

formula

${\displaystyle h_{K}={\frac {w_{K}\cdot {\sqrt {|D_{K}|}}}{2^{r_{1}}\cdot (2\pi )^{r_{2}}\cdot \operatorname {Reg} _{K}}}\lim _{s\to 1}(s-1)\zeta _{K}(s)}$

Here are

• ${\displaystyle r_{1}}$and the number of real and complex embeddings of ${\displaystyle 2r_{2}}$ ${\displaystyle K}$
• ${\displaystyle w_{K}}$the number of roots of unity in ${\displaystyle K}$
• ${\displaystyle \operatorname {Reg} _{K}}$the Dirichlet regulator of ${\displaystyle K}$
• ${\displaystyle D_{K}}$the discriminant of ${\displaystyle K}$
• ${\displaystyle \zeta _{K}(s)}$the Dedekind zeta function

Examples

The rational numbers

The number field of the rational numbers has a real and not a complex embedding, so . So the only roots of unity are . The Dirichlet regulator is the determinant of a matrix, that is , the discriminant and the trivial extension is . The Dedekind zeta function in this case is the Riemann zeta function . You get ${\displaystyle K=\mathbb {Q} }$ ${\displaystyle r_{1}=1,r_{2}=0}$ ${\displaystyle \pm 1}$ ${\displaystyle w_{\mathbb {Q} }=2}$ ${\displaystyle 0\times 0}$ ${\displaystyle \operatorname {Reg} _{\mathbb {Q} }=1}$ ${\displaystyle \mathbb {Q} /\mathbb {Q} }$ ${\displaystyle D_{\mathbb {Q} }=1}$ ${\displaystyle \zeta _{\mathbb {Q} }(s)}$ ${\displaystyle \zeta (s)}$

${\displaystyle h_{\mathbb {Q} }=\lim _{s\to 1}(s-1)\zeta (s)=1}$

in accordance with the known fact that is a main ideal ring . ${\displaystyle \mathbb {Z} }$

Imaginary square solids

For is and the residual of Dedekind's zeta function in is . One receives . ${\displaystyle K=\mathbb {Q} ({\sqrt {-5}})}$ ${\displaystyle r_{1}=0,r_{2}=1,w_{K}=2,\operatorname {Reg} _{K}=1,D_{K}=20}$ ${\displaystyle 1}$ ${\displaystyle {\frac {40\pi }{20^{3/2}}}}$ ${\displaystyle h_{K}=2}$

For is and shows a clever calculation of the residual of Dedekind's zeta function . ${\displaystyle K=\mathbb {Q} ({\sqrt {-163}})}$ ${\displaystyle r_{1}=0,r_{2}=1,w_{K}=2,\operatorname {Reg} _{K}=1,D_{K}=163}$ ${\displaystyle h_{K}=1}$

generalization

A generalization of the class number formula is the Lichtenbaum conjecture (named after Stephen Lichtenbaum ).

literature

• Analytic number theory. A tribute to Gauss and Dirichlet. Proceedings of the conference held in Göttingen, June 20-24, 2005. Edited by William Duke and Yuri Tschinkel. Clay Mathematics Proceedings, 7th American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. viii + 256 pp. ISBN 978-0-8218-4307-9
• Winfried Scharlau, Hans Opolka From Fermat to Minkowski. Lectures on the theory of numbers and its historic development , Springer Verlag, 1985 (Chapter 8: Dirichlet)

Web links

• Shurman: The dirichlet class number formula for imaginary quadratic fields
• Weston: Lectures on the dirichlet class number formula for imaginary quadratic fields