Ideal Class Group

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:

    “As to the family, I have never understood how that fits in with the other ideals—or, indeed, why it should be an ideal at all. A group of closely related persons living under one roof; it is a convenience, often a necessity, sometimes a pleasure, sometimes the reverse; but who first exalted it as admirable, an almost religious ideal?”
    —Rose Macaulay (1881–1958)

    “Where justice is denied, where poverty is enforced, where ignorance prevails, and where any one class is made to feel that society is in an organized conspiracy to oppress, rob, and degrade them, neither persons nor property will be safe.”
    —Frederick Douglass (c. 1817–1895)

    “He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
    —John Dos Passos (1896–1970)