A fractional ideal is a generalization of an ideal in a ring 
. Instead, a fractional ideal is contained in the number field 
, but has the property that there is an element 
such that
 |
(1)
|
is an ideal in 
. In particular, every element in 
can be written as a fraction, with a fixed denominator.
 |
(2)
|
Note that the multiplication of two fractional ideals is another fractional ideal.
For example, in the field 
, the set
 |
(3)
|
is a fractional ideal because
 |
(4)
|
Note that 
, where
 |
(5)
|
and so 
is an inverse to 
.
Given any fractional ideal 
there is always a fractional ideal 
such that 
. Consequently, the fractional ideals form an Abelian group by multiplication. The principal ideals generate a subgroup 
, and the quotient group is called the ideal class group.
