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
Famous quotes containing the words ideal, class and/or group:
“It is well worth the efforts of a lifetime to have attained knowledge which justifies an attack on the root of all evil—viz. the deadly atheism which asserts that because forms of evil have always existed in society, therefore they must always exist; and that the attainment of a high ideal is a hopeless chimera.”
—Elizabeth Blackwell (1821–1910)
“The class of citizens who provide at once their own food and their own raiment, may be viewed as the most truly independent and happy.”
—James Madison (1751–1836)
“The boys think they can all be athletes, and the girls think they can all be singers. That’s the way to fame and success. ...as a group blacks must give up their illusions.”
—Kristin Hunter (b. 1931)