Ideal class group

The ideal class group is a concept from the mathematical branch of algebraic number theory. It is a measure of how far the whole ring is removed in an algebraic number field of owning unique prime factorization. Your order is called the class number.

Definition ( for Dedekind rings)

It should be a Dedekind ring with quotient field, such as is the ring in an algebraic number field. Then the ideal class group is defined as the quotient group

It is

• The group of fractional ideals, that is, of finitely generated submodules of which contain not only the zero, with the product

• The subgroup of principal fractional ideals, that is, the submodules of the form

In the case of number fields to write mostly for.

The equivalence classes of the quotient group can be explicitly described as follows: Two broken ideals and are equivalent if there is an element, such that.

Properties

• Is the whole ring of an algebraic number field, so is finite.

Related terms

For an algebraic number field, there is an extension, the ( small ) between Hilbert class field. The Galois group is canonically isomorphic to the ideal class group, and every ideal of is in a principal ideal.