It is now time to prove the reciprocity law, the primary result in class field theory. I know I haven’t posted on this topic in a little while, so new readers (if they don’t already know this material) may want to review the strategy of the proof and the meaning of the Artin lemma (which is useful in reducing this to the cyclotomic case).
1. The cyclic reciprocity law
Well, I’ve already stated it before multiple times, but here it is:
Theorem 1 (Reciprocity law, cyclic case) Let
be a cyclic extension of number fields of degree
. Then the reciprocity law holds for
: there is an admissible cycle
such that the kernel of the map
is
, and the Artin map consequently induces an isomorphism
The proof of this theorem is a little sly and devious.
Recall that, for any admissible cycle , we have
by the conjunction of the first and second inequalities, and the Artin map is surjective. If we prove that the kernel of the Artin map is contained in
, then we’ll be done by the obvious count.
This is what we shall do.
Let be a fractional ideal prime to
which is in the kernel of the Artin map. We will prove that
in a slow, careful way. The first step is to factor
. We will apply Artin’s lemma for each prime
, leading to a lattice
where each splits completely in
, and
is cyclotomic, contained in
for some suitable
. We can choose the
successively so that
is not divisible by any of the primes dividing
.
Lemma 2 The field
, the compositum of all the
, satisfies
.
This is a fairly straightforward consequence of Galois theory. OK, cool. So we still need to do the reduction though.
1.1. The ideal
We can choose , an ideal of
, such that
for
a generator. We can do this by the surjectivity of the Artin map, and moreover so that
is prime to the primes above
. Then
satisfies
too. This ideal
is going to be a generic bookkeeping device with which we adjust
to get something in
.
1.2. Bookkeeping
Suppose . Then if
is a prime of
prolonging
, we have
as well by complete splitting. In particular, we have
but since is cyclotomic, we have the reciprocity law, and this means we can write
for very close to 1 at all the primes ramified in
, and
not divisible by those primes.
So, let’s take the norm down to ; then
But now is close to 1 at the primes of
, i.e. in
, if
was chosen properly (i.e. REALLY close to 1 at those primes, which is kosher in view of the reciprocity law for cyclotomic extensions). Moreover
is a norm from
.
The proof is now almost over. We have represented as a power of a fixed ideal times something in
.
1.3. Putting everything together
In particular, we have is
times some norm times something close to 1 at relevant primes. Taking the product, we see that
is
times some norm times something close to 1 at relevant primes. But now
because this is equal to
, so that sum is divisible by
, and
is a norm. We have proved that
is in the
subgroup, and this completes the proof.
2. The general reciprocity law
Again, it wouldn’t be appropriately dramatic if I didn’t state it in a box, right?
Theorem 3 (Artin reciprocity) For a finite abelian extension
of number fields, the Artin map has a conductor
and so factors into an isomorphism
The proof is now surprisingly easy. We can represent as a compositum
of cyclic extensions over
, by writing
as an intersection of subgroups
such that
is cyclic and taking the fixed fields
of
. The compositum
corresponds by Galois theory to the intersection
of
, and
is cyclic.
Now, there exist cycles such that each
maps to 1 in
by the reciprocity law (and existence of conductor) for cyclic extensions. Taking the product
, we find a conductor
such that the Artin map on
acts trivially on each
, hence on
.
Thus is a conductor for the Artin symbol on
, so the Artin map factors through
and since, first of all, it is surjective; and second of all, the former group is of order at most that of the latter (by the second inequality), we have that the Artin map is an isomorphism.
The proof of the reciprocity law, the prime result of class field theory, is now complete.
Next time, we will take stock of how far we have come, discuss corollaries, and plan for the future.
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