Kummer Theory of Number Fields

Assume now that the group of th roots of unity is contained in . Using Galois cohomology we obtain a relatively simple classification of all abelian extensions of with Galois group cyclic of order dividing . Moreover, since the action of on is trivial, by our hypothesis that , we see that

One can prove via calculations with discriminants, etc. that is unramified outside and and the primes that divide . Moreover, and this is a much bigger result, one can combine this with facts about class groups and unit groups to prove the following theorem:

We first argue that we can enlarge so that the ring

all

is a principal ideal domain.
Note that for any , the ring is a Dedekind domain.
Also, the condition
means that in the prime ideal factorization of the fractional ideal
, we have that
occurs to a nonnegative power. Thus we are
allowing denominators at the primes in . Since the class group of
is finite, there are primes
that generate
the class group as a group (for example, take all primes with norm up to
the Minkowski bound). Enlarge to contain the primes
.
Note that the ideal
is the unit ideal (we have
for some ; then
,
so
is the unit ideal, hence
is the unit ideal by unique factorization in the Dedekind
domain .)
Then is a principal ideal domain, since every ideal
of is equivalent modulo a principal ideal
to a product of ideals
. Note that we have used
that Next enlarge so that all primes over are in . Note that is still a PID. Let

all

Then a refinement of the arguments at the beginning of
this section show that is generated by all th roots
of the elements of . It thus sufficies to prove
that is finite.
There is a natural map

Recall that we proved *Dirichlet's unit theorem* (see
Theorem 8.1.2), which asserts that the group is a
finitely generated abelian group of rank . More generally, we
now show that
is a finitely generated abelian group of
rank
. Once we have shown this, then since is torsion
group that is a quotient of a finitely generated group, we will conclude
that is finite,
which will prove the theorem.

Thus it remains to prove that has rank . Let be the primes in . Define a map by

William Stein 2012-09-24