In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group.

= History and origin of the ideal class group =

The first ideal class groups encountered in , as was recognised at the time.

Later ).

Somewhat later again s The answer comes in the form all of them, if and only if the ideal class group (which is a finite group) has just one element .

= Technical development =

If R is an integral domain, define a relation (mathematics) ~ on nonzero ideals of R by I ~ J whenever there exist nonzero elements a and b of R such that ( a ) I = ( b ) J . (Here the notation ( a ) means the principal ideal of R consisting of all the multiples of a .) It is easily shown that this is an equivalence relation. The equivalence classes are called the ideal classes of R . Ideal classes can be multiplied: if [ I ] denotes the equivalence class of the ideal I , then the multiplication [ I ][ J ] = [ IJ ] is well-defined and commutative. The principal ideals form the ideal class [ R ] which serves as an identity element for this multiplication.

If R is a ring of algebraic integers, or more generally a Dedekind domain, the multiplication defined above turns the set of ideal classes into an abelian group, the ideal class group of R . The group property of existence of inverse elements is not immediate, but follows from an alternative route of development (see fractional ideal).

The ideal class group is trivial (i.e. contains only its identity element) if and only if all ideals of R are principal. In this sense, the ideal class group measures how far R is from being a principal ideal domain, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domains if and only if they are principal ideal domains).

The number of ideal classes (the class number of R ) may be infinite in general. But if R is in fact a ring of algebraic integers, then this number is always finite. This is one of the main results of classical algebraic number theory.

Computation of the class group is hard, in general; it can be done by hand for algebraic number fields of small Discriminant, using a theorem of Hermann Minkowski. This result gives a bound on the norm of an ideal in a particular class. In general the bound is not sharp enough to make the calculation practical for fields with large discriminant, but computers are well suited to the task.

It was remarked above that the ideal class group provides part of the answer to the question of how much ideals behave like ring elements in a Dedekind domain. The other part of the answer is provided by the multiplicative group (mathematics) of units of the Dedekind domain (and this is the rest of the reason for introducing the concept of fractional ideal, as well).

Define a map from K {0} to the set of all nonzero fractional ideals of R by sending every element to the principal (fractional) ideal it generates. This is a group homomorphism; its kernel of a homomorphism is the group of units of R , and its cokernel is the ideal class group of R . The failure of these groups to be trivial is a precise measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.

The mapping from rings of integers R to their corresponding class groups is functorial, and the class group can be subsumed under the heading of algebraic K-theory, with K0( R ) being the functor assigning to R its ideal class group; more precisely, K_0(R) = Bbb{Z} imes C(R), where C ( R ) is the class group. Higher K groups can also be employed and interpreted arithmetically in connection to rings of integers.

= Examples of ideal class groups =

The rings Z, Z[ i ], and Z[ w ], (where i is a square root of -1 and w is a cube root of 1) are all principal ideal domains, and so have class number 1: that is, they have trivial ideal class groups. If k is a field, then the polynomial ring k [ X 1, X 2, X 3, ...] is an integral domain. It has a countably infinite set of ideal classes.

If d is a .

If, on the other hand, d > 0, then it is unknown whether there are infinitely many fields Q( √ d ) with class number 1. Computational results indicate that there are a great many such fields to say the least.

= Connections to class field theory =

Class field theory is a branch of algebraic number field which seeks to classify all the Galois theorys of a given algebraic number field, meaning Galois extensions with abelian Galois group. A particularly beautiful example is found in the Hilbert class field of a number field, which can be defined as the maximal unramified abelian extension of such a field. The Hilbert class field L of a number field K is unique and has the following properties:

Neither property is particularly easy to prove.

== See also ==