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:
“We have reason to be grateful for celestial phenomena, for they chiefly answer to the ideal in man.”
—Henry David Thoreau (1817–1862)
“The traveler to the United States will do well ... to prepare himself for the class-consciousness of the natives. This differs from the already familiar English version in being more extreme and based more firmly on the conviction that the class to which the speaker belongs is inherently superior to all others.”
—John Kenneth Galbraith (b. 1908)
“Once it was a boat, quite wooden
and with no business, no salt water under it
and in need of some paint. It was no more
than a group of boards. But you hoisted her, rigged her.
She’s been elected.”
—Anne Sexton (1928–1974)