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Notes on class field theory

Kiran S. Kedlaya

Section 6.3 Adèles and idèles in field extensions

Reference.

[37], VI.1 and VI.2.
Up to now, we have considered the ring of adèles associated to a single number field. We now turn to the effect of a field extension on this construction.

Subsection Adèles in field extensions

Definition 6.3.1.

If \(L/K\) is an extension of number fields, we get an embedding \(\AA_K \hookrightarrow \AA_L\) as follows: given \(\alpha \in \AA_K\text{,}\) each place \(w\) of \(L\) restricts to a place \(v\) of \(K\text{,}\) so it makes sense to declare that the \(w\)-component of the image of \(\alpha\) shall equal \(\alpha_v\text{.}\) This embedding induces an inclusion \(I_K \hookrightarrow I_L\) of idèle groups.
All automorphisms of \(L/K\) act naturally on on \(\AA_L\) and \(I_L\text{.}\) More generally, if \(g \in \Gal(\overline{K}/K)\text{,}\) then \(g\) maps \(L\) to some other extension \(L^g\) of \(K\text{,}\) and \(g\) induces isomorphisms
\begin{equation*} \AA_L \to \AA_{L^g}, \qquad I_L \cong I_{L^g}, \qquad C_L \to C_{L^g}. \end{equation*}
Explicitly, if \((\alpha_w)_w \in \AA_L\) and \(g \in G\text{,}\) then \(g\) maps the completion \(L_w\) of \(L\) to a completion \(L_{w^g}\) of \(L^g\text{.}\) (Remember that a place \(w\) of \(L\) corresponds to an absolute value \(|\cdot|_w\) on \(L\text{;}\) the absolute value \(|\cdot|_{w^g}\) on \(L^g\) is given by \(|a^g|_{w^g} = |a|_w\text{.}\))

Remark 6.3.2.

A more conceptual interpretation of the previous discussion is to identify \(\AA_L\) with the tensor product \(\AA_K \otimes_K L\text{.}\) In particular, this is a good way to see the Galois action on \(\AA_L\text{.}\) See Exercise 1.

Remark 6.3.3.

When \(K\) is totally real, it is possible to show that every automorphism of \(\AA_K\) is induced by an automorphism of \(K\) over \(\QQ\text{,}\) even if we ignore topology and consider automorphisms of the underlying ring which need not be continuous. See Exercise 4. This breaks down when \(K\) has complex places because \(\CC\) has many automorphisms as a field without topology: the automorphism group acts transitively on \(\CC \setminus \overline{\QQ}\text{.}\)

Subsection Trace and norm

Definition 6.3.4.

For \(L/K\) an extension of number fields, we define the trace map \(\Trace_{L/K}: \AA_L \to \AA_K\) and the norm map \(\Norm_{L/K}: I_L \to I_K\) by the formulas
\begin{equation*} \Trace_{L/K}(x) = \sum_g x^g, \qquad \Norm_{L/K}(x) = \prod_g x^g \end{equation*}
where \(g\) runs over coset representatives of \(\Gal(\overline{K}/L)\) in \(\Gal(\overline{K}/K)\text{.}\) Here the sum and product take place in the adèle and idèle rings of the Galois closure of \(L\) over \(K\text{;}\) in particular, if \(L/K\) is Galois, \(g\) simply runs over \(\Gal(L/K)\) and the arithmetic takes place in \(\AA_L\text{.}\)
In terms of components, these definitions translate as
\begin{align*} (\Trace_{L/K}(\alpha))_{v} \amp = \sum_{w | v} \Trace_{L_w/K_v}(\alpha_w)\\ (\Norm_{L/K}(\alpha))_{v} \amp = \prod_{w | v} \Norm_{L_w/K_v}(\alpha_w). \end{align*}
The trace and norm as defined here are compatible with the usual definitions for principal adèles/idèles. In particular, the norm induces a map \(\Norm_{L/K}: C_L \to C_K\text{.}\)

Remark 6.3.5.

You can also define the trace of an adèle \(\alpha \in \AA_L\) as the trace of addition by \(\alpha\) as an endomorphism of the finite free \(\AA_K\)-module \(\AA_L\text{,}\) and the norm of an idèle \(\alpha \in I_L\) as the determinant of multiplication by \(\alpha\) as an automorphism of the finite free \(\AA_K\)-module \(\AA_L\text{.}\) (Yes, the action is on the adèles in both cases. Remember from Remark 6.2.3 that idèles should be thought of as automorphisms of the adèles, not as elements of the adèle ring, in order to topologize them correctly.)

Subsection Idèle groups and class groups

Proof.

If \(v\) is a place of \(K\text{,}\) then for each place \(w\) of \(K\) above \(v\text{,}\) the decomposition group \(G_w\) of \(w\) is isomorphic to \(\Gal(L_w/K_v)\text{.}\) So if \((\alpha)\) is an adèle or idèle which is \(G\)-invariant, then \(\alpha_w\) is \(\Gal(L_w/K_v)\)-invariant for each \(w\text{,}\) so belongs to \(K_v\text{.}\) Moreover, \(G\) acts transitively on the places \(w\) above \(v\text{,}\) so \(\alpha_w = \alpha_{w'}\) for any two places \(w, w'\) above \(v\text{.}\) Thus \((\alpha)\) is an adèle or idèle over \(K\text{.}\)
This has the following consequence for idèle class groups. Note that for \(L/K\) any extension of number fields, we can see that \(C_K \to C_L\) is injective from the fact that \(L^* \cap I_K = K^*\) within \(I_L\text{,}\) which follows from the fact that \(L \cap \AA_K = K\) within \(\AA_L\) (e.g., by Exercise 2).

Proof.

We start with an exact sequence
\begin{equation*} 1 \to L^* \to I_L \to C_L \to 1 \end{equation*}
of \(G\)-modules. Taking \(G\)-invariants, we get a long exact sequence
\begin{equation*} 1 \to (L^*)^G = K^* \to (I_L)^G = I_K \to C_L^G \to H^1(G, L^*)\text{,} \end{equation*}
and the last term is 1 by Theorem 90 (Lemma 1.2.3). So we again have a short exact sequence, and \(C_L^G \cong I_K/K^* = C_K\text{.}\)

Remark 6.3.8.

There is no analogue of Corollary 6.3.7 for ideal class groups: the map \(\Cl(K) \to \Cl(L)^G\) is neither injective nor surjective in general (Exercise 5). This is our first hint of why the idèle class group will be a more convenient target for a reciprocity map than the ideal class group.
The group \(\Cl(L)^G\) is classically known as the group of ambiguous classes of \(L/K\text{.}\) This is related to the concept of a Pólya field from Remark 2.3.13; see [5].

Exercises Exercises

1.

Let \(L/K\) be a finite extension of number fields. Prove that the natural map \(\AA_K \otimes_K L \to \AA_L\) is an isomorphism. In other words, if \(\alpha_1,\dots,\alpha_n\) form a basis of \(L\) as a \(K\)-vector space, then they also form a basis of \(\AA_L\) as an \(\AA_K\)-module.
Hint.
Show first that for any place \(v\) of \(K\text{,}\) any basis of \(L\) as a \(K\)-vector space also forms a basis of \(\prod_{w|v} L_w\) as a \(K_v\)-vector space.

2.

Let \(K\) be a number field. Prove that the integral closure of \(\QQ\) in \(\AA_K\) is equal to \(K\text{.}\)
Hint.
Suppose to the contrary that the integral closure contains some larger number field \(L\text{.}\) By Corollary 2.4.12, there are infinitely many primes of \(K\) which do not split in \(L\text{;}\) use one of these to obtain a contradiction.

3.

Prove the following converse to Exercise 2: if \(L/K\) is an extension of number fields such that \(K + \AA_{L,S} = \AA_L\) for some finite set of places \(S\) of \(L\text{,}\) prove that \(K=L\text{.}\)
Hint.
Use the fact that there are infinitely many primes of \(K\) that do not split completely in \(L\) (Corollary 2.4.12).

4.

Let \(K\) be a totally real Galois number field. Prove that the automorphism group of \(\AA_K\) as a bare ring (ignoring its topology) equals \(\Gal(K/\QQ)\text{.}\)
Hint.
By Exercise 2, any automorphism of \(\AA_K\) acts on \(K\text{.}\) We thus have homomorphisms \(\Gal(K/\QQ) \to \Aut(\AA_K) \to \Gal(K/\QQ)\) whose composition is the identity; it thus remains to check that \(\Aut(\AA_K) \to \Gal(K/\QQ)\) is injective. For this, apply Exercise 5 as in Exercise 6.

5.

Let \(L/K\) be a Galois extension of number fields with Galois group \(G\text{.}\)
  1. Give an example for which \(\Cl(K) \to \Cl(L)^G\) fails to be injective.
  2. Give an example for which \(\Cl(K) \to \Cl(L)^G\) fails to be surjective.
Hint.
One way to produce failures of injectivity is via the principal ideal theorem (Theorem 2.3.1). One way to produce failures of surjectivity is to find quadratic fields with class group \(\ZZ/4\ZZ\text{.}\)
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