using Minkowski’s constant to find a class number
We will use the theorem of Minkowski (see the parent entry (http://planetmath.org/MinkowskisConstant)).
Theorem (Minkowski’s Theorem).
so in the three cases:
Now, suppose that is an arbitrary class in . By the theorem, there exists an ideal , representative of , such that:
and therefore . Since the only ideal of norm one is the trivial ideal , which is principal, the class is also the trivial class in . Hence there is only one class in the class group, and the class number is one for the three fields and .
Let . The discriminant is and the Minkowski’s bound reads:
Suppose that is an arbitrary class in . By the theorem, there exists an ideal , representative of , such that:
and therefore or . However,
so the ideal is split in and the prime ideals
are the only ones of norm . Since they are principal, the class is the trivial class, and the class group is trivial. Hence, the class number of is one.
|Title||using Minkowski’s constant to find a class number|
|Date of creation||2013-03-22 15:05:38|
|Last modified on||2013-03-22 15:05:38|
|Last modified by||alozano (2414)|