Editing Ideal number (section)

==Example==
For instance, let <math>y</math> be a [[root of a polynomial|root]] of <math>y^2 + y + 6 = 0</math>, then the ring of integers of the field <math>\mathbb{Q}(y)</math> is <math>\mathbb{Z}[y]</math>, which means all <math>a + b \cdot y</math> with <math>a</math> and <math>b</math> integers form the ring of integers. An example of a nonprincipal ideal in this ring is the set of all <math>2 a + y \cdot b</math> where <math>a</math> and <math>b</math> are 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 <math>w</math> satisfying <math>w^3 - w - 1 = 0</math> to <math>\mathbb{Q}(y)</math>, giving <math>\mathbb{Q}(y,w)</math>. An ideal number for the nonprincipal ideal <math>2 a + y \cdot b</math> is <math>\iota = (-8-16y-18w+12w^2+10yw+yw^2)/23</math>. Since this satisfies the equation
<math>\iota^6-2\iota^5+13\iota^4-15\iota^3+16\iota^2+28\iota+8 = 0</math> it is an algebraic integer.

All elements of the ring of integers of the class field which when multiplied by <math>\iota</math> give a result in <math>\mathbb{Z}[y]</math> are of the form <math>a \cdot \alpha + y \cdot \beta</math>, where

:<math>\alpha = (-7+9y-33w-24w^2+3yw-2yw^2)/23</math>

and

:<math>\beta = (-27-8y-9w+6w^2-18yw-11yw^2)/23.</math>

The coefficients α and β are also algebraic integers, satisfying

:<math>\alpha^6+7\alpha^5+8\alpha^4-15\alpha^3+26\alpha^2-8\alpha+8=0</math>

and

:<math>\beta^6+4\beta^5+35\beta^4+112\beta^3+162\beta^2+108\beta+27=0</math>

respectively. Multiplying <math>a \cdot \alpha + b \cdot \beta</math> by the ideal number <math>\iota</math> gives <math>2 a + b \cdot y</math>, which is the nonprincipal ideal.