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 . Since this satisfies the equation 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
and
The coefficients α and β are also algebraic integers, satisfying
and
respectively. Multiplying aα + bβ by the ideal number ι gives 2a + by, which is the nonprincipal ideal.
Read more about this topic: Ideal Number
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)