In this series of posts I shall try to sketch (at least statements of) the basic results of global class field theory, following John Tate’s notes from the 1966 Brighton Conference. I begin by emphasising that I am writing them mainly in order to try to learn the material myself rather than really as an exposition to help others. As such, I expect to make countless errors, which expert readers are encouraged to point out.

It was conjectured by Hilbert that there exists, for a number field ${K}$ living inside a fixed algebraic closure, a unique totally unramified Galois extension ${L/K}$ with ${Gal(L/K)}$ isomorphic to the ideal class group of ${K}$. Such an extension is obviously abelian, and in fact Hilbert conjectured further that it is the maximal totally unramified abelian extension of ${K}$. Of course, Hilbert was right, and proving his conjecture is one of our main tasks.

In general, we will be studying a field ${K}$ which is usually a number field (or sometimes the function field of a curve over a finite field). Our overall goal seems to be to come up with a theory relating certain groups connected with the ideal class group of ${K}$ to the Galois groups of extensions of ${K}$. More specifically, we will define the idele group ${J_K}$ of ${K}$, which fits into an exact sequence

$\displaystyle 0 \rightarrow J_K^{S_\infty} \rightarrow J_K \rightarrow I_K \rightarrow 0,$

where ${I_K}$ is the group of fractional ideals and where ${J_K^{S_\infty}}$ is the subgroup of ideles all of whose prime factors’ correspond to archimedean valuations. In some sense therefore, we want to imagine archimedean valuations as primes at infinity’ and then ideles are a way to allow ideals to be divisible by these primes. With these defined, our main objects of study will be Artin maps

$\displaystyle \psi_{L/K}: J_K \rightarrow Gal(L/K),$

which give information about the splitting of primes in ${L/K}$ and provide the desired link between ideal class groups and the Galois groups.

In this first post we take the first tentative steps towards defining the Artin map, but will make no further mention of ideles yet. Let us set up some terminology. Fix ${K}$ a global field (a finite extension of ${{\mathbb Q}}$ or some ${\mathbb{F}_q(T)}$). A prime of ${K}$ is defined to be an equivalence class of nontrivial valuations on ${K}$. Primes ${v}$ are either discrete (they correspond to a prime ideal ${\mathfrak{p}_v = \{x| v(x) \geq 1\}}$, by satisfying an ultrametric law) or archimedean. We let ${M_K}$ denote the set of all primes of ${K}$, ${S^\infty = S^\infty_k}$ the set of archimedean primes and ${M_K^*}$ the set of discrete primes.

Now take ${L/K}$ a finite abelian extension, and let ${M_K^u}$ be the subset of ${M_K^*}$ consisting of unramified discrete primes. Our goal in this post will be to find a natural map

$\displaystyle F_{L/K}: M_k^u\rightarrow Gal(L/K),$

which we will extend in future posts to the Artin map ${\psi_{L/K}}$. To begin, let us start with some basic facts about how ${G=Gal(L/K)}$ acts on ${M_k}$ and the corresponding local fields.

We define the action of ${G}$ on ${M_L}$ by ${|\sigma x|_{\sigma w} := |x|_w}$. With this definition we get a left action ${\sigma(\tau w) = (\sigma \tau) w}$, and if ${v}$ is discrete it corresponds to the obvious action on prime ideals

$\displaystyle \mathfrak{p}_{\sigma w} = \{x| \sigma w(x) \geq 1\} = \{\sigma(x)| w(x) \geq 1 \} = \sigma \mathfrak{p}_w,$

and similarly ${\sigma}$ moves the local rings around in a natural fashion.

Now, there is a natural restriction map ${M_L \rightarrow M_K}$, and since ${\sigma(x) = x}$ for all ${x \in K}$, ${|x|_{\sigma w} = |x|_w}$ for all ${x \in K, \sigma \in G}$. Therefore the image of a Galois orbit in ${M_L}$ is just a single element of ${M_K}$. In fact we will show that ${G}$ acts transitively on the primes over a fixed ${v \in M_K}$, so since any such ${v}$ can be extended to ${L}$ in at least one way, the restriction map induces a correspondence:

$\displaystyle \{ G\text{-orbits of } M_L \} \leftrightarrow M_k.$

How can we prove this transitivity? Well, let ${G_w = \{\sigma \in G| \sigma w = w\}}$ be the decomposition group of ${w}$. The orbit of ${w}$ will simply be ${w=w_1, w_2,...,w_r}$ where ${r=(G:G_w)}$, with one prime ${w_i}$ corresponding to each coset ${\sigma_i G_w}$. For each ${w_i}$, an element of ${G_w}$ gives an isometric automorphism, which extends to an automorphism of the completion ${L_{w_i} \rightarrow L_{w_i}}$ and thus we have a natural embedding

$\displaystyle G_w \hookrightarrow Aut(L_{w_i}/K_v).$

Hence we get, by basic field theory, the bounds

$\displaystyle |G| = r|G_w| \leq \sum_{i=1}^r |Aut(L_{w_i}/K_v)| \leq \sum_{i=1}^r [L_{w_i}: K_v]$

Suppose there are some other primes ${w_{r+1},...,w_s}$ also extending ${v}$. Crucially, the natural map ${L \otimes_K K_v \rightarrow \prod_{i=1}^s L_{w_i}}$ must be surjective, as an application of the weak approximation theorem. This implies that

$\displaystyle \sum_{i=1}^s [L_{w_i}:K_v] \leq [L:K] = |G|.$

Putting our two estimates together, we get that equality holds at every stage, so in particular ${s=r}$ (the action is transitive).

Note we have also proved that ${L_w/K_v}$ is Galois and has Galois group naturally isomorphic to ${G_w}$. In fact, this group is about to play a key role in another capacity. Under the additional assumptions that ${w}$ is discrete and unramified over ${v}$, this Galois group is canonically isomorphic to the Galois group of residue fields ${Gal(k(w)/k(v))}$ (where ${k(v) = \mathcal{O}_v/\mathfrak{p}_v}$). So we have canonical isomorphisms

$\displaystyle G \supseteq G_w \cong Gal(L_w/K_v) \cong Gal(k(w)/k(v)).$

But the residue fields are just finite fields, so letting ${q=|k(v)|}$, ${Gal(k(w)/k(v))}$ is generated by ${Frob_q: x \mapsto x^q}$. Since the above isomorphisms are canonical, this corresponds to a canonical generating element ${\sigma_W}$ of ${G_w}$, which is as such called the Frobenius automorphism associated with ${w}$, characterised by the property ${\sigma_w(a) \equiv a^q (\text{mod }\mathfrak{p}_w)}$ for all ${a \in L}$.

Wait! What just happened? We took a prime ${w}$ of ${M_L^*}$ (assumed not to be ramified) and its decomposition group contained a special element ${\sigma_w}$. In other words, we defined a natural mapping ${M_L^* \rightarrow Gal(L/K)}$. Recall that we were looking for a map ${M_K^u \rightarrow Gal(L/K)}$, so it feels like we are almost done.

But just a minute ago we proved that the Galois orbits of ${M_L}$ are in bijective correspondence with ${M_K}$, and this correspondence obviously respects the properties of being discrete and unramified, so the obvious thing to hope is that ${\sigma_w = \sigma_{\tau w}}$ for all ${\tau \in G}$, since we can then just define ${\sigma_v}$ to be this element (the Frobenius automorphism associated with every prime above ${v}$).

Oh, but this is obvious, because ${Gal(L/K)}$ is abelian, and by the characterising property of ${\sigma_w}$, ${\sigma_{\tau w}}$ must be conjugate to it. So we are done. Provided ${L/K}$ is abelian, we have a lovely well-defined map called `take the associated Frobenius automorphism of any prime above ${v}$

$\displaystyle F_{L/K}: M_K^u \rightarrow Gal(L/K).$

Next time, we will extend this to a group homomorphism and state the Artin reciprocity law, our first deep theorem which tells us a lot about how these maps work. However, before we finish it is worth remarking how the above map relates to the splitting of an unramified prime ${v}$ in ${L}$.

Indeed, we saw in the above discussion about the action of ${G}$ on ${M_L}$ that the set of primes ${w}$ dividing ${v}$ is isomorphic to ${(G:G_w)}$. Now, ${G_w}$ is actually generated by ${\sigma_w}$, and hence by the symmetry of the action, we know how ${v}$ splits. If ${\sigma_w}$ has order ${f}$, ${v}$ must split into ${|G|/f}$ distinct factors, each of degree ${f}$. So knowing ${\sigma_v = F_{L/K}(v)}$ tells us how ${v}$ splits and gives us a generator for the decomposition group. This is an very nice situation to be in, and hints at some of the wide-reaching consequences of the theory.