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:

    I’m no idealist to believe firmly in the integrity of our courts and in the jury system—that is no ideal to me, it is a living, working reality. Gentlemen, a court is no better than each man of you sitting before me on this jury. A court is only as sound as its jury, and a jury is only as sound as the men who make it up.
    Harper Lee (b. 1926)

    God ... created a number of possibilities in case some of his prototypes failed—that is the meaning of evolution.
    Graham Greene (1904–1991)