Let and be integers. Consider the ring
Consider now the ring
is divisible by both and .
It is also a well-ordered group. Hence it is generated by a single positive element , called the least common multiple (LCM) of and .
, and are subings of . Moreover, and . So by the Chinese remainder theorem,
which can be written as
In general, let . As before, we can define
is divisible by
Then and . As before,
So , i.e.
If we replace by for any number field , then takes the form
Taking cardinalities gives the following equation in terms of ideal norms