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.