Ideal Class Group


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 GroupHistory 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 ...
Dedekind Domain - Fractional Ideals and The Class Group
... 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 ...
Takagi Existence Theorem - Earlier Work
... 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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