Ideal Number

In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This means that there is an element of the ring of integers of the Hilbert class field, which is an ideal number, such that the original nonprincipal ideal is equal to the collection of all multiples of this ideal number by elements of this ring of integers that lie in the original field's ring of integers.

Read more about Ideal Number:  Example, History

Famous quotes containing the words ideal and/or number:

    The ideal has many names, and beauty is but one of them.
    W. Somerset Maugham (1874–1965)

    Not too many years ago, a child’s experience was limited by how far he or she could ride a bicycle or by the physical boundaries that parents set. Today ... the real boundaries of a child’s life are set more by the number of available cable channels and videotapes, by the simulated reality of videogames, by the number of megabytes of memory in the home computer. Now kids can go anywhere, as long as they stay inside the electronic bubble.
    Richard Louv (20th century)