Ideal class group

The ideal class group is a term from the mathematical branch of algebraic number theory . It is a measure of how far the integral ring in an algebraic number field is from having a unique prime factorization . Their order is called the class number .

Definition (for dedekind rings)

Let it be a Dedekind ring with a quotient field , for example the wholeness ring in an algebraic number field . Then the ideal class group is defined as the factor group ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle \operatorname {Pic} A}$

${\displaystyle \operatorname {Pic} A=J_{A}/P_{A}.}$

It is

• ${\displaystyle J_{A}}$the group of broken ideals , d. H. of the finitely generated -submodules of , which do not only contain the zero, with the product ${\displaystyle A}$ ${\displaystyle K}$

${\displaystyle IJ=\left\{\left.\sum _{i=1}^{n}a_{i}b_{i}\right|a_{i}\in I,b_{i}\in J\right\},}$

The group is the free Abelian group on the prime ideals of . ${\displaystyle J_{A}}$ ${\displaystyle A}$

• ${\displaystyle P_{A}}$the subgroup of broken main ideals, d. H. the sub-modules of the form

${\displaystyle (a)=A\cdot a\subset K}$

for . ${\displaystyle a\in K^{\times }}$

In the case of number fields, one usually writes for . ${\displaystyle \operatorname {Cl} _{K}}$ ${\displaystyle \operatorname {Pic} A}$

The equivalence classes of the factor group can also be explicitly described as follows: Two broken ideals and are equivalent if there is an element such that . ${\displaystyle I}$ ${\displaystyle J}$ ${\displaystyle \lambda \in K^{\times }}$ ${\displaystyle I=\lambda J}$

properties

• ${\displaystyle \operatorname {Pic} A}$is trivial if and only then, d. H. the class number is 1 when is a major ideal ring , and that is equivalent to having a unique prime factorization . ${\displaystyle A}$ ${\displaystyle A}$
• If the totality ring is an algebraic number field , then is finite. ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle \operatorname {Cl} _{K}}$
• The algebraic K-theory provides a generalization of the concept of the ideal class group . If is the integral ring of an algebraic number field , then is . ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle \operatorname {K} _{0}(A)=\mathbb {Z} \oplus \operatorname {Cl} _{K}}$
• The class number formula puts the class number of a number field in connection with the residual of its Dedekind zeta function in . ${\displaystyle 1}$

Examples

Let it be a quadratic number field , i.e. H. for a square-free number . ${\displaystyle K}$ ${\displaystyle K=\mathbb {Q} ({\sqrt {d}})}$ ${\displaystyle d\in \mathbb {Z} }$

The only negative square-free numbers for which the ideal class group of is trivial are ${\displaystyle d<0}$ ${\displaystyle K=\mathbb {Q} ({\sqrt {d}})}$

${\displaystyle d=-1,-2,-3,-7,-11,-19,-43,-67,-163.}$

This was suspected by Carl Friedrich Gauss and proved by Kurt Heegner in 1952 ; Heegner's proof, however, only found recognition after a work published by Harold Stark in 1967 .

It is not known whether there are infinitely many positive square-free numbers for which the ideal class group of is trivial, but there are many calculated examples of this. ${\displaystyle d>0}$ ${\displaystyle K=\mathbb {Q} ({\sqrt {d}})}$

Related terms

There is an extension for an algebraic number field , the (small) Hilbert class field . The Galois group is canonically isomorphic to the ideal class group, and every ideal of becomes a main ideal in. ${\displaystyle K}$ ${\displaystyle H/K}$ ${\displaystyle \operatorname {Gal} (H/K)}$ ${\displaystyle K}$ ${\displaystyle H}$

literature

• Jürgen Neukirch : Algebraic number theory . Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 .