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
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
by the conjunction of the first and second inequalities, and the Artin map
This is what we shall do. (more…)
a prime of 
unramified in 
. Then we can find a field 
, a subextension of 
, with 
such that in the lattice of fields 
, so that 
is cyclotomic 
such that it is divisible only by arbitrarily large primes. 
of 
of 
, 
) such that 
is cyclotomic and a given prime 
. From this, we shall be able to deduce the Artin law for cyclic extensions, whence the general case will be a “mere” corollary.
. Then there exists 
, prime to 
, such that 
has order divisible by 
of order divisible by 
have trivial intersection.
, we can assume that 
.
. It is also necessarily surjective (a consequence of the first inequality). We don’t know that it factors through 
, however.
, which implies that the map must be an isomorphism, since 
is sufficieintly close to 1 at a large set of primes, then the ideal 
has trivial Artin symbol. This is what we need to prove.
for 
a subgroup of 
such that 
, in particular 
of primes such that:
for 
the ![{[M:k]}](https://s0.wp.com/latex.php?latex=%7B%5BM%3Ak%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
whose degree we can easily compute; it is
and 
and prove that 
. The whole proof is too long for one blog post, so I will do step 1 (as well as some preliminary index computations—yes, these are quite fun—today). 
. Here 
means that the two differ by a bounded quantity as 
. 
is asymptotically 
for 
, i.e. an analog of the standard prime number theorem. In number fields. We actually need a slight refinement thereof.
is a character of the group 
, we have
, and otherwise it tends either to a finite limit or 
. 
. There is a generalization of this to number fields, called the Dedekind zeta function. The Dedekind-zeta function is not defined by summing over 
for 
in the ring of integers (minus 0). Why not? Because the ring of integers is not a unique factorization domain in general, and therefore we don’t get a nice product formula. 
of fractional ideals prime to the elements of 
.
be a prime in 
of 
are the rings of integers in 
, respectively, then we have a field extension 
. As is well-known, there is a surjective homomoprhism of the decomposition group 
of 
whose kernel, called the inertia group, is of degree 
.
is cyclic, because these are finite fields. The Galois group is generated by a canonically determined Frobenius element which sends 
. We can lift this to an element 
of 
.
. In particular,
acts on the ideles 
, and also on 
(clearly). As a result, we get an action on the idele classes 
. There is a map 
; I claim that it is an injection, and the fixed points of 
are precisely the points of 
. This can be proved using group cohomology. We have an exact sequence 
, and consequently one has a long exact sequence
is the ordinary (non-Tate) cohomology groups, that is to say just the 
and 
, we find that 
, q.e.d.
. It in particular implies that 
, and since this group is none other than
of 
, though—this is the unit theorem. This isomorphism is by the log map, and it is even a 
.
-free module of finite rank on which 
, and choose a lattice in 
be two lattices in 
. ![{\mathbb{Q}[G]}](https://s0.wp.com/latex.php?latex=%7B%5Cmathbb%7BQ%7D%5BG%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
-modules 
, and 
-modules 
defined similarly. Moreover, we have![\displaystyle \mathbb{R} \otimes_{{\mathbb Q}} \hom_{\mathbb{Q}[G]}( L_{{\mathbb Q}}, L_{{\mathbb Q}}) \simeq \hom_{{\mathbb R}[G]}(L_{{\mathbb R}}, L'_{{\mathbb R}}).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cmathbb%7BR%7D+%5Cotimes_%7B%7B%5Cmathbb+Q%7D%7D+%5Chom_%7B%5Cmathbb%7BQ%7D%5BG%5D%7D%28+L_%7B%7B%5Cmathbb+Q%7D%7D%2C+L_%7B%7B%5Cmathbb+Q%7D%7D%29+%5Csimeq+%5Chom_%7B%7B%5Cmathbb+R%7D%5BG%5D%7D%28L_%7B%7B%5Cmathbb+R%7D%7D%2C+L%27_%7B%7B%5Cmathbb+R%7D%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002)
Climbing Mount Bourbaki 