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:

    ... probably all of the women in this book are working to make part of the same quilt to keep us from freezing to death in a world that grows harsher and bleaker—where male is the norm and the ideal human being is hard, violent and cold: a macho rock. Every woman who makes of her living something strong and good is sharing bread with us.
    Marge Piercy (b. 1936)

    He has more to impart than to receive from his generation. He is another such a strong and finished workman in his craft as Samuel Johnson was, and, like him, makes the literary class respectable.
    Henry David Thoreau (1817–1862)

    Instead of seeing society as a collection of clearly defined “interest groups,” society must be reconceptualized as a complex network of groups of interacting individuals whose membership and communication patterns are seldom confined to one such group alone.
    Diana Crane (b. 1933)