**Ideal Class Group**

In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field (or more generally any Dedekind domain) can be described by a certain group known as an **ideal class group** (or **class group**). If this group is finite (as it is in the case of the ring of integers of a number field), then the order of the group is called the **class number**. The multiplicative theory of a Dedekind domain is intimately tied to the structure of its class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain.

Read more about Ideal Class Group: History and Origin of The Ideal Class Group, Definition, Properties, Relation With The Group of Units, Examples of Ideal Class Groups, Connections To Class Field Theory

### Other articles related to "ideal class group, class, ideal, ideals":

**Ideal Class Group**- Connections To Class Field Theory

...

**Class**field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois ... 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 ... The Hilbert

**class**field L of a number field K is unique and has the following properties Every

**ideal**of the ring of integers of K becomes principal in L, i.e ...

... A fractional

**ideal**is a nonzero R-submodule I of K for which there exists a nonzero x in K such that (We remark that this is not exactly the same as the ... integral

**ideals**— fractional

**ideals**.) Given two fractional

**ideals**I and J, one defines their product IJ as the set of all finite sums the product IJ is again a ... The set Frac(R) of all fractional

**ideals**endowed with the above product is a commutative semigroup and in fact a monoid the identity element is the fractional

**ideal**R ...

... In this case the generalized

**ideal class group**is the

**ideal class group**of K, and the existence theorem says there exists a unique abelian extension L/K with ... This extension is called the Hilbert

**class**field ... A further and special property of the Hilbert

**class**field, not true of other abelian extensions of a number field, is that all

**ideals**in a number field become ...

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