Editing Ideal number (section)

{{Short description|Algebraic Integer which represents an ideal in the ring of integers of a number field}}
In [[number theory]], an '''ideal number''' is an [[algebraic integer]] which represents an [[ideal (ring theory)|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 [[ideal (ring theory)|ideal]]s 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. By the [[principal ideal theorem]], any non-principal 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 non-principal 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.