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 while trying to solve Fermat's last theorem, 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.

Example

For instance, let y be a root of y² + y + 6 = 0, then the ring of integers of the field is , which means all a + by with a and b integers form the ring of integers. An example of a nonprincipal ideal in this ring is 2a + yb with a and b integers; the cube of this ideal is principal, and in fact the class group is cyclic of order three. The corresponding class field is obtained by adjoining an element w satisfying w³ - w - 1 = 0 to , giving . An ideal number for the nonprincipal ideal 2a + yb is ι = ( - 8 - 16y - 18w + 12w² + 10yw + yw²) / 23. Since this satisfies the equation ι⁶ - 2ι⁵ + 13ι⁴ - 15ι³ + 16ι² + 28ι + 8 = 0 it is an algebraic integer.

All elements of the ring of integers of the class field which when multiplied by ι give a result in are of the form aα+bβ, where α = ( - 7 + 9y - 33w - 24w² + 3yw - 2yw²) / 23 and β = ( - 27 - 8y - 9w + 6w² - 18yw - 11yw²) / 23. The coefficients α and β are also algebraic integers, satisfying α⁶ + 7α⁵ + 8α⁴ - 15α³ + 26α² - 8α + 8 = 0 and β⁶ + 4β⁵ + 35β⁴ + 112β³ + 162β² + 108β + 27 = 0 respectively. Multiplying aα + bβ by the ideal number ι gives 2a + by, which is the nonprincipal ideal.