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Section 7.4 The existence theorem

Reference.

[36] VII.6, VII.9, [37] VI.4, VI.6.

With the “abstract” reciprocity theorem in hand, we now prove the existence theorem in its idelic formulation (see Theorem 6.4.2). Modulo the pending reconciliation of Artin reciprocity with abstract reciprocity (see Proposition 7.5.7), this will imply the classical version of the existence theorem: every generalized ideal class group of a number field is identified by Artin reciprocity with the Galois group of a suitable abelian extension (Theorem 2.2.8).

As in the proof of the local existence theorem (Theorem 4.3.11), having access to the (abstract) reciprocity law and the norm limitation theorem reduces the task of proving the existence theorem to the “topological” assertion that every open subgroup of \(C_L\) of finite index contains a norm subgroup. For this, we can essentially rerun the Kummer-theoretic argument from the local case.

We then give the closely related algebraic proof of the Second Inequality (Theorem 7.2.10).

Subsection A base case for the existence theorem

As in the proof of the local existence theorem (Theorem 4.3.11), the key to the proof of Theorem 7.4.8 is showing that for any given number field \(K\text{,}\) we can find finite extensions \(L/K\) for which the groups \(\Norm_{L/K} C_L\) can be made arbitrarily small. In preparation for an inductive proof, we establish a key base case using Kummer theory.

Let \(J\) be the preimage of \(U\) under the projection \(I_K \to C_K\text{,}\) so that \(J\) is open in \(I_K\) of finite index. Then \(J\) contains a subgroup of the form

\begin{equation*} V = \prod_{v \in S} \{1\} \times \prod_{v \notin S} \gotho_{K_v}^* \end{equation*}

for some finite set \(S\) of places of \(K\) containing the infinite places, which by Corollary 6.2.10 we may choose large enough so that \(K^* I_{K,S} = I_K\text{.}\) The group \(J\) must also contain \(I_K^p\text{,}\) and hence \(W_S\text{.}\)

We continue with a lemma that allows to detect whether certain elements of a number field are \(p\)-th powers based on whether this happens locally. This amounts to a carefully chosen special case of the Grunwald-Wang theorem (Remark 7.2.13).

It is clear that

\begin{equation*} (\gotho_{K,S}^*)^p \subseteq W_S \cap K^*\text{.} \end{equation*}

To prove the reverse inclusion, note that for any \(y \in W_S \cap K^*\text{,}\) if we set \(L = K(y^{1/p})\text{,}\) then every place \(v \in S\) is split in \(L\) and every place \(v \notin S\) is unramified in \(L\text{,}\) yielding

\begin{equation*} \Norm_{L/K} I_{L,S} = I_{K,S}. \end{equation*}

Since \(I_K = K^* I_{K,S}\text{,}\) this implies \(\Norm_{L/K} C_L = C_K\text{.}\) By the First Inequality (Theorem 7.1.2), this implies \(L = K\) and so \(y \in (K^*)^p\text{.}\)

This will in turn enable us to compute the norm group for a certain compositum of Kummer extensions.

By Corollary 6.2.11 and the assumption that \(K\) contains a primitive \(p\)-th root of unity, the group \(\gotho_{K,S}^*/(\gotho_{K,S}^*)^p\) is finite of order \(p^s\text{,}\) yielding \([L:K] = p^s\text{.}\) By local reciprocity (Lemma 4.3.8), we have \(K^* W_S/K^* \subseteq \Norm_{L/K} C_L\text{;}\) to prove equality, it will suffice to check that these groups have the same index in \(C_K\text{.}\)

Consider now the exact sequence

\begin{equation*} 1 \to \frac{\gotho_{K,S}^*}{\gotho_{K,S}^* \cap W_S} \to \frac{I_{K,S}}{W_S} \to \frac{C_K}{K^* W_S/K^*} \to 1. \end{equation*}

By Lemma 7.4.2, the group on the left has order \([\gotho_{K,S}^*:(\gotho_{K,S}^*)^p] = p^s\text{.}\) By Lemma 7.4.4, the group in the middle has order \(p^{2s}\text{.}\) Thus using the abstract global reciprocity isomorphism (Theorem 7.3.8), we obtain

\begin{equation*} [C_K:K^* W_S/K^*] = p^s = [L:K] = [C_K:\Norm_{L/K} C_L], \end{equation*}

as desired.

Here is the local calculation used in the proof of Lemma 7.4.3.

We separate cases as follows.

  1. If \(v\) is a real place, then \(p=2\text{,}\) \(p^2/|p|_v = 2\text{,}\) and

    \begin{equation*} K_v^*/(K_v^*)^p = \RR^*/(\RR^*)^2 = \RR^*/\RR^+ \cong \ZZ/2\ZZ. \end{equation*}

  2. If \(v\) is a complex place, then \(p^2/|p|_v\) = 1 according to our conventions (Definition 6.1.10), and \(K_v^*/(K_v^*)^p\) is trivial because \(\CC^*\) is \(p\)-divisible.

  3. If \(v\) is a finite place not lying above \(p\text{,}\) then \(p^2/|p|_v = p^2\) and \(K_v^*/(K_v^*)^p\) is generated by \(\zeta_p\) and a uniformizer of \(K_v\text{.}\)

  4. If \(v\) is a finite place above \(p\text{,}\) then \(|p|_v = p^{-n}\) for some positive integer \(n\text{,}\) so \(p^2/|p|_v = p^{n+2}\text{.}\) Since \(K_v^* \cong \gotho_{K_v}^* \times \ZZ\text{,}\) it suffices to check that \([\gotho_{K_v}^*: (\gotho_{K_v}^*)^p] = p^{n+1}\text{.}\) For this, see Exercise 1.

We finally put everything together to get a key special case of the existence theorem.

Subsection Proof of the existence theorem

Building on the base case offered by Lemma 7.4.5, we now finish the proof of the existence theorem.

Take \(K' = K(\zeta_p)\text{.}\) Let \(U'\) be the inverse image of \(U\) in \(C_{K'}\text{.}\) By Theorem 7.3.8, \([C_K:\Norm_{K'/K} C_{K'}] = [K':K]\) is coprime to \(p\text{;}\) consequently, \([C_{K'}:U'] = p\text{.}\) By Lemma 7.4.5, there exists a finite extension \(L/K'\) such that \(\Norm_{L/K'} C_{L} \subseteq U'\text{;}\) then \(\Norm_{L/K} C_L \subseteq \Norm_{K'/K} U' \subseteq U\text{.}\)

We proceed by induction on the index \([C_K:U]\text{,}\) with Lemma 7.4.6 as the base case. Otherwise, choose an intermediate subgroup \(V\) between \(U\) and \(C_K\text{.}\) By the induction hypothesis, \(V\) contains \(N = \Norm_{L/K} C_L\) for some finite extension \(L\) of \(K\text{.}\) Then

\begin{equation*} [N : (U \cap N)] = [UN : U] \leq [V:U]. \end{equation*}

Let \(W\) be the subgroup of \(C_L\) consisting of those \(x\) whose norms lie in \(U\text{.}\) Then

\begin{equation*} [C_L:W] \leq [N:U \cap N] \leq [V:U], \end{equation*}

so by the induction hypothesis \(W\) contains \(\Norm_{M/L} C_M\) for some finite extension \(M/L\text{.}\) Thus \(U\) contains \(\Norm_{M/K} C_M\text{,}\) as desired.

For any finite abelian extension \(L/K\text{,}\) \(\Norm_{L/K} C_L\) is a subgroup of \(C_K\) which is open (by Remark 7.1.7) of index \([L:K]\) (by Theorem 7.3.8). Moreover, by Corollary 5.3.13, the correspondence \(L \mapsto \Norm_{L/K} C_L\) is injective.

Conversely, let \(U\) be an open subgroup of \(C_K\) of finite index. By Lemma 7.4.7, there exists a finite extension \(L_1/K\) such that \(\Norm_{L_1/K} C_{L_1} \subseteq U\text{.}\) By the adelic norm limitation theorem (Theorem 6.4.3), we also have \(\Norm_{L_1/K} C_{L_1} = \Norm_{L_2/K} C_{L_2} \subseteq U\) for \(L_2/K\) the maximal abelian subextension of \(L_1/K\text{.}\) By Theorem 7.3.8 again, we have an isomorphism \(\Gal(L_2/K) \cong C_K/\Norm_{L_2/K} C_{L_2}\text{,}\) via which the subgroup \(U/\Norm_{L_2/K} C_{L_2}\) corresponds to a subgroup \(H\) of \(\Gal(L_2/K)\text{.}\) Taking \(L\) to be the fixed field of \(H\text{,}\) we deduce that \(\Norm_{L/K} C_L = U\) as desired.

Remark 7.4.9.

As with the proof of the local existence theorem, the proof of Theorem 7.4.8 is constructive in principle but not in practice: it involves constructing some extension much larger than the desired abelian extension, then invoking the norm limitation theorem to get down to an abelian extension. We remind the reader that there is no easy fix known for this (Remark 2.2.10).

Subsection An algebraic approach to the Second Inequality

Drawing inspiration from the calculation of norm groups given in Lemma 7.4.3, we now explain how to use similar ideas to give an algebraic proof of the Second Inequality. Again, the key case is where \(L/K\) is a cyclic extension of number fields of prime degree \(p\) and \(\zeta_p \in K\text{.}\) To modify the calculation from Lemma 7.4.3 to compute the norm group of a single Kummer extension, we use a second set of places.

By Kummer theory (Theorem 1.2.6), we can choose a finite set \(S\) of places of \(K\) containing all infinite places, all places that ramify in \(L\text{,}\) and all places above \(p\) so that \(L = K(\Delta^{1/p})\) for \(\Delta = \gotho_{K,S}^* \cap (L^*)^p\text{.}\) This remains true after enlarging \(S\text{,}\) so by Corollary 6.2.10 we can further ensure that \(I_K = I_{K,S} K^*\text{.}\)

Put \(N = K((\gotho_{K,S}^*)^{1/p})\text{.}\) By Kummer theory again

\begin{equation*} \Gal(N/K) \cong \Hom(\gotho_{K,S}^*/(\gotho_{K,S}^*)^p, \ZZ/p\ZZ). \end{equation*}

By Corollary 6.2.11, \(\gotho_{K,S}^*/(\gotho_{K,S}^*)^p \cong (\ZZ/p\ZZ)^s\text{.}\) Choose generators \(g_1, \dots, g_{s-1}\) of \(\Gal(N/L)\text{;}\) these correspond in \(\Hom(\gotho_{K,S}^*/(\gotho_{K,S}^*)^p, \ZZ/p\ZZ)\) to a set of homomorphisms whose common kernel is precisely \(\Delta/(\gotho_{K,S}^*)^p\text{.}\) We thus need to find, for each \(g_i\text{,}\) a place \(v_i\) such that the kernel of \(g_i\) is the same as the kernel of \(\gotho_{K,S}^* \to K_{v_i}^*/(K_{v_i}^*)^p\text{;}\) we can then take \(T = \{v_1,\dots,v_{s-1}\}\text{.}\)

Let \(N_i\) be the fixed field of \(g_i\text{;}\) by Corollary 7.1.16 (which we deduced from the First Inequality), there are infinitely many primes of \(N_i\) that do not split in \(N\text{.}\) So we can choose a place \(w_i\) of each \(N_i\) such that their restrictions \(v_i\) to \(K\) are distinct, not contained in \(S\text{,}\) and don't divide \(p\text{.}\)

We claim \(N_i\) is the maximal subextension of \(N/K\) in which \(v_i\) splits completely (i.e., the decomposition field of \(v_i\)). On one hand, \(v_i\) does not split completely in \(N\text{,}\) so the decomposition field is no larger than \(N_i\text{.}\) On the other hand, the decomposition field is the fixed field of the decomposition group, which has exponent \(p\) and is cyclic (since \(v_i\) does not ramify in \(N\)). Thus it must have index \(p\) in \(N\text{,}\) so must be \(N_i\) itself.

Thus \(L = \bigcap N_i\) is the maximal subextension of \(N\) in which all of the \(v_i\) split completely. We conclude that for \(x \in \gotho_{K,S}^*\text{,}\) \(x\) belongs to \(\Delta\) iff \(K_{v_i}(x^{1/p}) = K_{v_i}\) for all \(i\text{,}\) which occurs iff \(x \in K_{v_i}^p\text{.}\) That is, \(\Delta\) is precisely the kernel of the map \(\gotho_{K,S}^* \to \prod_i K_{v_i}^*/(K_{v_i}^*)^p\text{.}\) This proves the claim.

We have the following modified version of Lemma 7.4.2.

It is again clear that

\begin{equation*} (\gotho_{K,S \cup T}^*)^p \subseteq W_{S,T} \cap K^*\text{.} \end{equation*}

To prove the reverse inclusion, it will again suffice to prove that \(y \in W_{S, T} \cap K^*\text{,}\) if we set \(L = K(y^{1/p})\text{,}\) then \(\Norm_{L/K} C_L = C_K\text{;}\) namely, Theorem 7.1.2 will then imply \(L = K\) and so \(y \in (K^*)^p\text{.}\)

Since \(\gotho_{K,S}^* \to \prod_{v \in T} \gotho_{K_v}^*/(\gotho_{K_v}^*)^p\) is surjective, any element of \(I_{K,S \cup bigT}\) can be written as the product of an element of \(\gotho_{K,S}^*\) with an element of \(I_{K,S \cup T}\) which is a \(p\)-th power at each place of \(T\text{.}\) In particular, by Lemma 4.3.8 such an element is a norm from \(L\) at each place of \(T\text{;}\) we can now reprise the proof of Lemma 7.4.2, skipping over the places in \(T\text{,}\) to deduce that we have a norm from \(L\text{.}\)

By Lemma 7.4.11, we have an exact sequence

\begin{equation*} 1 \to \frac{\gotho_{K,S\cup T}^*}{(\gotho_{K,S \cup T}^*)^p} \to \frac{I_{K,S \cup T}}{W_{S,T}} \to \frac{C_K}{K^* W_{S,T}/K^*} \to 1, \end{equation*}

with which we may compute as in Lemma 7.4.3: the left group has order \(p^{\#(S \cup T)} = p^{2s-1}\) by Corollary 6.2.11 while the middle group has order \(p^{2s}\) by Lemma 7.4.4, so

\begin{equation*} [C_K: K^* W_{S,T}/K^*] = p. \end{equation*}

Meanwhile, we can check by local reciprocity that \(\Norm_{L/K} I_{L,S} = I_{K,S}\) (compare the proof of Lemma 7.4.11).

  • For \(v \in S\text{,}\) elements of \((K_v^*)^p\) are norms from any abelian extension of \(K_v\) of exponent \(p\) (by Lemma 4.3.8).

  • For \(v \in T\text{,}\) \(v\) splits in \(L\) and so \(L_w = K_v\text{.}\)

  • For \(v \notin S \cup T\text{,}\) \(v\) is unramified in \(L\) and so \(\Norm_{L_w/K_v} \gotho_{L_w}^* = \gotho_{K_v}^*\text{.}\)

Dence \(K^* W_{S,T} \subseteq \Norm_{L/K} C_L\text{,}\) completing the proof.

For \(x \in C_K\text{,}\) \(\Norm_{L/K} (x) = x^p\text{;}\) this implies that both groups in question are killed by \(p\text{.}\) In particular, multiplication by \(d = [K':K]\text{,}\) which divides \(p-1\text{,}\) is an isomorphism on these groups.

Suppose \(x \in C_K\) maps to the identity in \(H^0_T(\Gal(L'/K'), C_{L'})\text{.}\) We can then choose a representative of class of \(x\) in \(H^0_T(\Gal(L/K), C_L)\) of the form \(y^d\text{;}\) then \(y\) also maps to the identity in \(H^0_T(\Gal(L'/K'), C_{L'})\text{.}\) That is, \(y = \Norm_{L'/K'}(z')\) for some \(z' \in C_{L'}\text{,}\) and

\begin{equation*} y^d = \Norm_{K'/K}(y) = \Norm_{L'/K}(z') \in \Norm_{L/K} C_L. \end{equation*}

Thus \(x \in \Norm_{L/K} C_L\text{,}\) as needed.

As in the proof of Theorem 7.2.10, we use an induction argument to reduce to proving that for \(L/K\) cyclic,

\begin{equation*} H^0_T(\Gal(L/K), C_L) \leq [L:K]. \end{equation*}

In fact, the same induction (considering \(H^2\) in place of \(H^0_T\)) allows us to further reduce to the case where \([L:K] = p\) is prime.

Let \(K' = K(\zeta_p)\) and \(L' = L(\zeta_p)\text{;}\) then \(K'\) and \(L\) are linearly disjoint over \(K\) (since their degrees are coprime), so \([L':K'] = [L:K] = p\) and the Galois groups of \(L/K\) and \(L'/K'\) are canonically isomorphic. By Lemma 7.4.12 and Lemma 7.4.13,

\begin{equation*} \#H^0_T(\Gal(L/K), C_L) \leq \#H^0_T(\Gal(L'/K'), C_{L'}) \leq [L':K'] = p \end{equation*}

as desired.

Exercises Exercises

1.

Complete the proof of Lemma 7.4.4 by showing that if \(|p|_v = p^{-n}\) for some positive integer \(n\text{,}\) then \([\gotho_{K_v}^*:(\gotho_{K_v}^*)^p] = p^{n+1}\text{.}\) (Remember that \(K\) is a number field containing a primitive \(p\)-th root of unity.)

Hint.

Using the logarithm map, we obtain an isomorphism \(\gotho_{K_v}^*/\mu_{K_v} \cong \ZZ_p^n\) (even when \(p=2\)). See [37], Proposition II.5.7 for details.

2.

Let \(K\) be a number field. Prove that for every positive integer \(n\text{,}\) \(C_K^n\) is the intersection of the norm groups \(\Norm_{L/K} C_L\) over all abelian extensions \(L/K\) of exponent \(n\text{.}\)