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:
“You’ll never succeed in idealizing hard work. Before you can dig mother earth you’ve got to take off your ideal jacket. The harder a man works, at brute labour, the thinner becomes his idealism, the darker his mind.”
—D.H. (David Herbert)
“You must drop all your democracy. You must not believe in “the people.” One class is no better than another. It must be a case of Wisdom, or Truth. Let the working classes be working classes. That is the truth. There must be an aristocracy of people who have wisdom, and there must be a Ruler: a Kaiser: no Presidents and democracies.”
—D.H. (David Herbert)
“Laughing at someone else is an excellent way of learning how to laugh at oneself; and questioning what seem to be the absurd beliefs of another group is a good way of recognizing the potential absurdity of many of one’s own cherished beliefs.”
—Gore Vidal (b. 1925)