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 equally impossible to forget our Friends, and to make them answer to our ideal. When they say farewell, then indeed we begin to keep them company. How often we find ourselves turning our backs on our actual Friends, that we may go and meet their ideal cousins.”
—Henry David Thoreau (1817–1862)
“Criminals are never very amusing. It’s because they’re failures. Those who make real money aren’t counted as criminals. This is a class distinction, not an ethical problem.”
—Orson Welles (1915–1985)
“A little group of wilful men reflecting no opinion but their own have rendered the great Government of the United States helpless and contemptible.”
—Woodrow Wilson (1856–1924)