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:
“Good friends, good books and a sleepy conscience: this is the ideal life.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“Planning ahead is a measure of class. The rich and even the middle class plan for future generations, but the poor can plan ahead only a few weeks or days.”
—Gloria Steinem (b. 1934)
“The conflict between the need to belong to a group and the need to be seen as unique and individual is the dominant struggle of adolescence.”
—Jeanne Elium (20th century)