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 coe