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:
“The contest between the Future and the Past is one between Divinity entering, and Divinity departing. You are welcome to try your experiments, and, if you can, to displace the actual order by that ideal republic you announce, of nothing but God will expel God.”
—Ralph Waldo Emerson (1803–1882)
“There is a certain class of people who prefer to say that their fathers came down in the world through their own follies than to boast that they rose in the world through their own industry and talents. It is the same shabby-genteel sentiment, the same vanity of birth which makes men prefer to believe that they are degenerated angels rather than elevated apes.”
—W. Winwood Reade (1838–1875)
“around our group I could hear the wilderness listen.”
—William Stafford (1914–1941)