Ideal number

In mathematics 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 Kummer, and lead to 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 Principalization theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This means there is an element of the ring of integers of the class field, which is an ideal number, such that all multiples times elements of this ring of integers lying in the ring of integers of the original field define the nonprincipal ideal.

Related Topics:
Mathematics - Algebraic integer - Number field - Kummer - Dedekind - Ideal - Ring - Principalization theorem - Hilbert class field - Ring of integers

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