Ideal number

From Wikipedia, the free encyclopedia
Jump to: navigation, search



For instance, let y be a root of y2 + y + 6 = 0, then the ring of integers of the field \Bbb{Q}(y) is \Bbb{Z}[y], 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 w3w − 1 = 0 to \Bbb{Q}(y), giving \Bbb{Q}(y,w). An ideal number for the nonprincipal ideal 2a + yb is \iota = (-8-16y-18w+12w^2+10yw+yw^2)/23. Since this satisfies the equation \iota^6-2\iota^5+13\iota^4-15\iota^3+16\iota^2+28\iota+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 \Bbb{Z}[y] are of the form aα + bβ, where

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


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

The coefficients α and β are also algebraic integers, satisfying











External links[edit]