Ideal number: Encyclopedia II - Ideal number - Example

Ideal number - 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.

Adapted from the Wikipedia article "Example", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki