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Notes on class field theory

Kiran S. Kedlaya

Section 4.3 Local class field theory via Tate’s theorem

Reference.

[37] II.3, III.5. See also [1], Preliminaries, section 2 for a sketch of the proof of Tate’s theorem; note that the cohomology groups \(H^i\) considered therein are always Tate groups, which we notate as \(H^i_T\text{.}\)
For \(L/K\) a finite extension of local fields (of characteristic \(0\)), we have now computed that \(H^1(L/K) = 0\) (Lemma 1.2.3) and \(H^2(L/K)\) is cyclic of order \([L:K]\) (Proposition 4.2.1). We next use these ingredients to establish all of the statements of local class field theory.

Subsection Cup products in cohomology and Tate groups

We first describe cup products, which will give rise to the map in Tate’s theorem (Theorem 4.1.14). See also [37], Proposition II.1.38.

Proof.

We first treat the case of cohomology groups. Without loss of generality suppose that \(M\) is torsion-free. When \(i = 0\text{,}\) for any \(m \in H^0(G,M) = M^G\text{,}\) \(n \mapsto m \otimes n\) is a homomorphism of \(G\)-modules; we thus get maps \(H^j(G,N) \to H^j(G, M \otimes_\ZZ N)\) which are functorial in \(N\) and compatible with long exact sequences (here we are using that \(M\) is torsion-free and hence flat over \(\ZZ\)). For \(i \gt 0\text{,}\) we perform dimension shifting on \(M\) using the sequence
\begin{equation*} 0 \to M \to \Ind^G_1 \Res^G_1 M \to M' \to 0 \end{equation*}
from Proposition 3.2.6; recall that this sequence splits (Remark 3.2.7), so it remains exact upon tensoring with \(N\text{.}\) Note also that we need to check functoriality and compatibility with long exact sequences in \(M\text{;}\) we leave this to the reader.
For the Tate groups, we automatically have the claim when \(i,j \gt 0\) from the case of cohomology groups. We can then dimension shift downward, using sequences of the form
\begin{equation*} 0 \to M' \to \Ind^G_1 \Res^G_1 \to M \to 0 \end{equation*}
as in Proposition 3.2.6 (which splits at the level of \(\ZZ\)-modules), to deduce the claim for smaller indices.

Remark 4.3.2.

See [37], Proposition I.1.39 for more properties of the cup product, which are again proved by starting in degree 0 and then using dimension shifting. One of these is skew-symmetry:
\begin{equation*} x \cup y = (-1)^{ij} y \cup x \qquad (x \in H^i_T(G, M), y \in H^j_T(G,N))\text{.} \end{equation*}
Another is compatibility with restriction:
\begin{equation*} \Res(x \cup y) = \Res(x) \cup \Res(y)\text{.} \end{equation*}

Subsection Tate’s theorem

We next prove Tate’s theorem (Theorem 4.1.14). Note that right now, we only need this for solvable groups because every finite Galois extension of local fields has solvable Galois group (Remark 4.2.3); this allows for some simplification in the arguments. However, we will do the extra work to handle the general case for later use in the global setting.

Proof.

Let us first check that \(H^i_T(G,M) = 0\) for all \(i \geq 0\text{.}\) For \(G\) cyclic, this follows by periodicity (Theorem 3.4.1) from the hypothesis that \(H^1(G,M) = H^2(G,M) = 0\text{.}\) In fact this even yields vanishing for \(i \lt 0\text{,}\) but we will not be able to carry that through the following steps.
For \(G\) solvable, we prove that \(H^i_T(G,M) = 0\) for all \(i \geq 0\) by induction on \(\#G\) as follows. The trivial group serves as a trivial base case. Since \(G\) is solvable, if it is not the trivial group then it has a proper subgroup \(H\) for which \(G/H\) is cyclic. By the induction hypothesis, \(H^i_T(H,\Res^G_H M) = 0\) for all \(i \geq 0\text{.}\) Thus by the inflation-restriction exact sequence (Proposition 4.2.13),
\begin{equation*} 0 \to H^i(G/H, M^H) \to H^i(G, M) \to H^i(H, \Res^G_H M) \end{equation*}
is exact for all \(i \gt 0\text{;}\) the term on the end being zero, we have \(H^i(G/H, M^H) \cong H^i(G,M) = 0\) for \(i=1, 2\text{.}\) By periodicity again, \(H^i_T(G/H, M^H) = 0\) for all \(i \in \ZZ\text{,}\) so \(H^i(G/H, M^H) = 0\) for all \(i>0\text{,}\) and so \(H^i(G,M) = 0\) for \(i \gt 0\text{.}\) For \(i=0\text{,}\) the vanishing of \(H^0_T(G/H, M^H)\) means that every \(x \in M^G = (M^H)^{G/H}\) can be written in the form \(\Norm_{G/H}(y)\) for some \(y \in M^H\text{,}\) and the vanishing of \(H^0_T(H, \Res^G_H M)\) means that \(y\) can in turn be written in the form \(\Norm_H(z)\) for some \(z \in M\text{;}\) we thus have \(x = \Norm_G (z)\) and so \(H^0_T(G, M) = 0\text{.}\)
For \(G\) arbitrary, we prove that \(H^i_T(G,M) = 0\) for all \(i \geq 0\) by induction on \(\#G\) as follows. For \(i \geq 0\text{,}\) we already know that the group \(H^i_T(G,M)\) is killed by \(\#G\) (apply Example 3.2.24 for \(i \gt 0\) and check directly for \(i = 0\)), so it suffices to show that its \(p\)-primary component vanishes for any prime \(p\) dividing \(\#G\text{.}\) To check this, let \(G_p\) be any Sylow \(p\)-subgroup of \(G\text{.}\) For \(i \gt 0\text{,}\) as per Example 3.2.24 again, the composition
\begin{equation*} H^i(G, M) \stackrel{\Res}{\to} H^i(G_p, \Res^G_{G_p} M) \stackrel{\Cor}{\to} H^i(G, M) \end{equation*}
is multiplication by the integer \([G:G_p]\) which is prime to \(p\text{;}\) consequently, \(\Res\) induces an injective map on \(p\)-primary components. Since \(G_p\) is solvable and the \(G_p\)-module \(\Res^G_{G_p} M\) satisfies the vanishing hypothesis, we can apply the previous paragraph to deduce that \(H^i(G_p, \Res^G_{G_p} M) = 0\text{,}\) and hence \(H^i(G,M) = 0\text{.}\) For \(i=0\text{,}\) the vanishing of \(H^0_T(G_p, \Res^G_{G_p} M)\) means that every \(x \in M^G\) can be written in the form \(\Norm_{G_p}(y)\text{,}\) and so \([G:G_p]x = \Norm_G(y)\text{;}\) hence the \(p\)-primary part of \(H^0_T(G, M)\) also vanishes.
Finally, we treat the case \(i \lt 0\) by dimension shifting. We proceed by induction: we will show that if the claim holds for all \(i \geq -n\) for some nonnegative integer \(n\text{,}\) then the claim also holds for all \(i \geq -n-1\text{.}\) To this end, make the exact sequence
\begin{equation*} 0 \to N \to \Ind^G_1 \Res^G_1 M \to M \to 0 \end{equation*}
in which \(m \otimes [g]\) maps to \(m^g\) (from the proof of Proposition 3.2.6). For any subgroup \(H\) of \(G\text{,}\) the term in the middle is acyclic with respect to \(H\) (see Definition 3.2.4). From the long exact sequence, we obtain \(H^{i+1}_T(H, \Res^G_H N) \cong H^{i}_T(H, \Res^G_H M)\) for all \(i \in \ZZ\text{.}\) In particular, since we already know that \(H^i_T(H, \Res^G_H M) = 0\) for all \(i \geq -n\text{,}\) we deduce that \(H^i_T(H, \Res^G_H N) = 0\) for all \(i \geq -n+1\text{,}\) and in particular for \(i=1,2\text{.}\) We may thus apply the induction hypothesis to deduce that \(H^i_T(G, N) = 0\) for \(i \geq -n\text{.}\) Turning this around, we deduce that \(H^i_T(G, M) = 0\) for \(i \geq -n-1\text{.}\)

Proof.

Since \(\Cor \circ \Res = [G:H]\) (Example 3.2.24), \(\Res(\gamma)\) generates \(H^2(H,\Res^G_H M)\) for any \(H\text{.}\) We start out by explicitly constructing a \(G\)-module containing \(M\) in which \(\gamma\) becomes a coboundary.
Choose a 2-cocycle \(\phi\colon G^3 \to M\) representing \(\gamma\text{;}\) by the definition of a cocycle,
\begin{gather*} \phi(g_0 g, g_1 g, g_2 g) = \phi(g_0, g_1, g_2)^g\\ \phi(g_1, g_2, g_3) - \phi(g_0, g_2, g_3) + \phi(g_0, g_1, g_3) - \phi(g_0, g_1, g_2) = 0\text{.} \end{gather*}
Moreover, \(\phi\) is a coboundary if and only if it is of the form \(d(\rho)\text{,}\) that is, \(\phi(g_0, g_1, g_2) = \rho(g_1, g_2) - \rho(g_0, g_2) + \rho(g_0, g_1)\text{.}\) This \(\rho\) must itself be \(G\)-invariant: \(\rho(g_0, g_1)^g = \rho(g_0g, g_1g)\text{.}\) Thus \(\phi\) is a coboundary if and only if
\begin{equation*} \phi(e, g, hg) = \rho(e,h)^g - \rho(e,hg) + \rho(e,g)\text{.} \end{equation*}
Let \(M[\phi]\) be the direct sum of \(M\) with the free abelian group with one generator \(x_g\) for each element \(g\) of \(G - \{e\}\text{,}\) with the \(G\)-action
\begin{equation*} x_h^g = x_{hg} - x_g + \phi(e, g, hg)\text{.} \end{equation*}
(The symbol \(x_e\) should be interpreted as the element \(\phi(e,e,e)\) of \(M\text{.}\)) Using the cocycle property of \(\phi\text{,}\) one may verify that this is indeed a \(G\)-action; by construction, the cocycle \(\phi\) becomes zero in \(H^2(G, M[\phi])\) by setting \(\rho(e,g) = x_g\text{.}\) (Milne calls \(M[\phi]\) the splitting module of \(\phi\text{.}\)) Moreover, by the same token, for any \(H\text{,}\) the restriction of \(\phi\) to \(H\) also becomes zero in \(H^2(H, \Res^G_H M[\phi])\text{.}\)
The map \(\alpha\colon M[\phi] \to \ZZ[G]\) sending \(M\) to zero and \(x_g\) to \([g]-1\) is a homomorphism of \(G\)-modules. Actually it maps into the augmentation ideal \(I_G\text{,}\) and the sequence
\begin{equation*} 0 \to M \to M[\phi] \to I_G \to 0 \end{equation*}
is exact. Meanwhile, we also have the usual exact sequence
\begin{equation*} 0 \to I_G \to \ZZ[G] \to \ZZ \to 0\text{.} \end{equation*}
For any subgroup \(H\) of \(G\text{,}\) we can restrict to \(H\)-modules (which we leave out of the notation for brevity), then take the long exact sequence. In the second case, since \(\ZZ[G]\) is induced (for \(G\) and hence for \(H\)), its Tate groups all vanish and we get a dimension shift:
\begin{align*} H^1(H, I_G) &\cong H^0_T(H, \ZZ) = \ZZ/(\#H)\ZZ\\ H^2(H, I_G) &\cong H^1(H, \ZZ) = 0\text{.} \end{align*}
In the first case, we get
\begin{gather*} 0 = H^1(H, M) \to H^1(H, M[\phi]) \to H^1(H, I_G)\\ \to H^2(H, M) \to H^2(H, M[\phi]) \to H^2(H, I_G) = 0\text{.} \end{gather*}
Moreover, the map into \(H^2(H, M[\phi])\) vanishes by construction, so in fact \(H^2(H, M[\phi]) = 0\text{.}\) Now \(H^1(H, I_G) \to H^2(H,M)\) is a surjective map between two finite (even cyclic) groups of the same order \(\#H\text{,}\) so the map is also injective and \(H^1(H, M[\phi]) = 0\text{.}\)
We may now apply Lemma 4.3.3 to conclude that \(H^i_T(G, M[\phi]) = 0\) for all \(i\text{.}\) This allows us to use the four-term exact sequence
\begin{equation*} 0 \to M \to M[\phi] \to \ZZ[G] \to \ZZ \to 0 \end{equation*}
(as in the proof of Theorem 3.4.1) to obtain the dimension-shifting isomorphism \(H^i_T(G, \ZZ) \cong H^{i+2}_T(G, M)\text{.}\)

Remark 4.3.5.

The reader may detect a strong similarity between the proofs of Theorem 3.4.1 and Theorem 4.3.4. For a common generalization, see See [1], Preliminaries, section 2.

Remark 4.3.6.

In a similar vein, we can use cup products to give a different description of the periodicity isomorphism \(H^i_T(G,M) \to H^{i+2}(G,M)\) when \(G\) is cyclic: it is the cup product with the class in \(H^2_T(G, \ZZ)\) arising via the connecting homomorphism from the class in \(H^1_T(G, \QQ/\ZZ) = \Hom(G, \QQ/\ZZ)\) taking the chosen generator of \(G\) to \(\frac{1}{\#G}\text{.}\) Similarly, the inverse map is the cup product with the class in \(H^{-2}(G, \ZZ) \cong G\) corresponding to the chosen generator.

Subsection Local reciprocity isomorphisms and norm limitation

Remark 4.3.7.

Let \(L/K\) be a finite Galois extension of local fields. For any intermediate extension \(M/K\text{,}\) we know that \(H^1(L/M) = 0\) (by Lemma 1.2.3) and \(H^2(L/M)\) is cyclic of order \([L:M]\) (by Proposition 4.2.1). We may thus apply Theorem 4.3.4 with \(G = \Gal(L/K)\text{,}\) \(M = L^*\) to obtain isomorphisms
\begin{equation*} H^i_T(\Gal(L/K), \ZZ) \to H^{i+2}_T(L/K)\text{.} \end{equation*}
In particular, we get an isomorphism
\begin{equation} K^*/\Norm_{L/K} L^* = H^0_T(L/K) \to H^{-2}_T(\Gal(L/K), \ZZ) = \Gal(L/K)^{\ab}\text{.}\tag{4.3.1} \end{equation}
This immediately yields the the norm limitation theorem (Theorem 4.1.7): if \(M/K\) is the maximal abelian subextension of \(L/K\text{,}\) then \(\Norm_{L/K} L^* \subseteq \Norm_{M/K} M^*\) but both have the same finite index in \(K^*\text{,}\) and so must be equal.
To make the previous isomorphism canonical, we use the canonical isomorphism \(H^2(L/K) \cong \frac{1}{[L:K]}\ZZ/\ZZ\) from Proposition 4.2.1 to specify that our preferred generator of \(H^2(L/K)\) is the one corresponding to \(\frac{1}{[L:K]}\text{.}\) We thus deduce Theorem 4.1.11.

Remark 4.3.8.

From Remark 4.3.7, we immediately deduce the Galois case of the norm limitation theorem (Theorem 4.1.7): if \(L/K\) is a Galois extension of local fields and \(M/K\) is the maximal abelian subextension of \(L/K\text{,}\) then \(\Norm_{L/K} L^* \subseteq \Norm_{M/K} M^*\) but both have the same finite index in \(K^*\text{,}\) and so must be equal.
For \(L/K\) not necessarily Galois, we can reduce to the previous case as follows. The inclusion \(\Norm_{L/K} L^* \subseteq \Norm_{M/K} M^*\) remains evident, so we must check the reverse inclusion \(\Norm_{M/K} M^* \subseteq \Norm_{L/K} L^*\text{;}\) this may be checked after enlarging \(L\) in such a way as to preserve \(M\text{.}\) In particular we may replace \(L\) with its Galois closure, in which case the previous argument applies.

Remark 4.3.9.

As for the existence of the local reciprocity map (Theorem 4.1.2), we are not quite there yet: we have isomorphisms as in (4.3.1) for every \(L/K\text{,}\) but we still need to verify that the maps defined for various \(L\) over a fixed \(K\) fit together. Note that since we have the norm limitation theorem in hand, we need only check this compatibility for finite abelian extensions \(L/K\text{.}\) (Although this is not essential, we can further reduce to comparing \(L/K\) with its subextension \(M/K\) when both of them are cyclic, and even of prime power degree: see Exercise 1.) We also note that once this compatibility is established, part (a) of Theorem 4.1.2 will follow from the explicit computations in Section 4.2.
We address the issue by reformulating the local reciprocity map to avoid mixing cohomology and homology groups. For \(G\) abelian (not necessarily cyclic), we have canonical isomorphisms
\begin{align*} H^{-2}_T(G, \ZZ) &\cong G\\ H^0_T(G, \ZZ) &\cong \frac{1}{(\#G)}\ZZ /\ZZ\\ H^2_T(G, \ZZ) &\cong \Hom(G, \QQ/\ZZ) \end{align*}
and the cup product pairing
\begin{equation*} H^{-2}_T(G, \ZZ) \times H^2_T(G, \ZZ) \to H^0_T(G, \ZZ) \end{equation*}
is nondegenerate: any element in either group on the left that pairs to zero with everything is itself zero.
This means that we can give an alternate description of the local reciprocity isomorphism: the cup product pairing
\begin{equation*} H^0_T(L/K) \times H^2_T(\Gal(L/K), \ZZ) \to H^2_T(L/K) \stackrel{\inv_{L/K}}{\to} \QQ/\ZZ \end{equation*}
is also nondegenerate. If we precompose with the map \(H^0(L/K) \to H^0_T(L/K)\text{,}\) then everything is in terms of cohomology groups rather than Tate groups, so we can calculate more explicitly, as in Lemma 4.3.10. Even more importantly, we now have extended functoriality available (Example 3.2.24): if \(M/K\) is a subextension of \(L/K\text{,}\) then the cup product pairing is compatible with the restriction maps \(H^i(\Gal(L/K), *) \to H^i(\Gal(M/K), *)\) (see Remark 4.3.2), as are the invariant maps \(\inv_{L/K}\) and \(\inv_{M/K}\) (by Proposition 4.2.1). This allows us to conclude that the local reciprocity maps collate to a single map \(K^* \to \Gal(\overline{K}/K)^{\ab}\text{,}\) completing the proof of Theorem 4.1.2.
Alternatively, one can use the explicit proof of the existence theorem given by Lubin-Tate theory to make the calculation of the reciprocity map explicit enough to read off the compatibility with changing the extension. See Section 4.6.
We record an alternate formulation of the local reciprocity map that may be useful for some calculations.

Proof.

This amounts to carefully tracing through the dimension shifts we used in Proposition 4.3.1 to construct the cup product. For the details, see [48], XI.3, Proposition 2.
Using similar ideas, we can read off a compatibility of the local reciprocity map with changing the base field.

Proof.

Let \(M/K\) be an abelian extension of local fields. Then \(\Norm_{L/K}\) carries \(\Norm_{ML/L} (ML)^*\) to \(\Norm_{ML/K} (ML)^* \subseteq \Norm_{M/K} M^*\) and thus induces a map \(L^*/\Norm_{ML/L} (ML)^* \to K^*/\Norm_{M/K} K^*\text{;}\) the claim is that this is compatible with the natural map \(\Gal(ML/L) \to \Gal(M/K)\text{.}\)
For this, it is convenient to replace \(L^*\) with the group \((L \otimes_K M)^*\) and identify the latter with \(\Ind^G_H L^*\) for \(G = \Gal(M/K), H = \Gal(ML/L)\text{.}\) We can then build a diagram that looks like the one we want by applying ordinary functoriality for the map \(\Norm_{L/K}\colon (L \otimes_K M)^* \to M^*\) to the construction of Remark 4.3.7, translating back into the terms we want using Shapiro’s lemma (Lemma 3.2.3), and observing the relationship between the local invariant maps for \(K\) and \(L\) (as in the proof of Proposition 4.2.17).

Subsection The local existence theorem

We next address the local existence theorem (Theorem 4.1.5). This does not follow directly from cohomological considerations; instead we need to construct some extensions with small norm groups. Fortunately, since we have already established the norm limitation theorem, we do not need to construct abelian extensions; this will give us some flexibility.
We begin with a lemma, in which we take advantage of Kummer theory to establish a special case of the existence theorem.

Proof.

Let \(M\) be the compositum of all cyclic \(\ell\)-extensions of \(K\text{.}\) The group \(K^*/(K^*)^{\ell}\) is finite (Exercise 2), and hence is isomorphic to \((\ZZ/\ell \ZZ)^n\) for some positive integer \(n\text{.}\) By Kummer theory (Theorem 1.2.9), we also have \(\Gal(M/K) \cong (\ZZ/\ell \ZZ)^n\text{.}\) By the local reciprocity isomorphism (in the form obtained in Remark 4.3.7), \(K^*/\Norm_{M/K}M^* \cong (\ZZ/\ell \ZZ)^n\text{;}\) consequently, on one hand \((K^*)^{\ell} \subseteq \Norm_{M/K}M^*\text{,}\) and on other hand these subgroups of \(K^*\) have the same index \(\ell^n\text{.}\) They are thus equal, proving the claim.

Remark 4.3.14.

The conclusion of Lemma 4.3.13 remains true even if \(\ell\) is not prime; see Exercise 4. This statement can be interpreted in terms of the Hilbert symbol, whose properties generalize quadratic reciprocity to higher powers; see [37], III.4.
We deduce some corollaries of the existence theorem which are needed in its proof. (The arguments we give here depend squarely on characteristic \(0\text{;}\) some patching is needed in the positive characteristic case.)

Proof.

Apply Lemma 4.3.13 to the field \(K(\zeta_\ell)\) obtained by adjoining a primitive \(\ell\)-th root of unity to \(K\text{:}\) this tells us that \(x\) becomes an \(\ell\)-th power in \(K(\zeta_\ell)^*\text{.}\) By taking norms back to \(K\text{,}\) we deduce that \(x^{[K(\zeta_\ell):K]}\) is an \(\ell\)-th power in \(K^*\text{;}\) since \([K(\zeta_\ell):K]\) divides \(\ell-1\text{,}\) it is coprime to \(\ell\text{,}\) and so we deduce that \(x \in (K^*)^\ell\text{.}\)

Proof.

Let \(D_K\) be the intersection in question; note that \(D_K \subseteq \gotho_K^*\) by considering unramified extensions of \(K\text{,}\) so \(D_K\) is in particular a compact topological group. By Lemma 4.3.15, for every prime \(\ell\text{,}\) every element of \(D_K\) is the \(\ell\)-th power of an element of \(D_K\) we will show that in fact every element of \(D_K\) is the \(\ell\)-th power of an element of \(D_K\) itself. This will show that \(D_K\) is a divisible abelian group, and in particular every element is an \(n\)-th power for every positive integer \(n\text{;}\) it will then follow from Exercise 3 that \(D_K\) is the trivial group. (Alternatively, one can follow the suggestion of Remark 4.3.14 and prove that the conclusion of Lemma 4.3.13 retains true when \(\ell\) is replaced by an arbitrary positive integer \(n\text{,}\) and then apply Exercise 3 directly.)
We first need to verify something which might seem obvious but isn’t quite: for \(L/K\) a finite extension,
\begin{equation*} \Norm_{L/K} D_L = D_K\text{.} \end{equation*}
This isn’t obvious because for \(x \in D_K\text{,}\) for each individual finite extension \(M\) of \(K\) we can write \(x = \Norm_{M/K}(z)\) for some \(z \in M^*\text{,}\) but it is not apparent that we can force the elements \(y = \Norm_{M/L}(z)\) to all be equal. It is nonetheless true because, for any given \(M\) the set of such \(y\) is a nonempty compact subset of \(U_L\text{,}\) and any finite intersection of these subsets is nonempty (because we can pass to a large enough field to contain all of the \(M\) in question and bring an element from there); so the whole intersection is nonempty.
Now let \(\ell\) be a prime and choose \(x \in D_K\text{.}\) For each finite extension \(L\) of \(K\) containing a primitive \(\ell\)-th root of unity, let \(E(L)\) be the set of \(\ell\)-th roots of \(x\) in \(K\) which belong to \(\Norm_{L/K} L^*\text{.}\) This set is finite (of cardinality at most \(\ell\)) and nonempty: we have \(x = \Norm_{L/K}(y)\) for some \(y \in D_L\) by the previous paragraph, so \(y\) has an \(\ell\)-th root \(z\) in \(L\) and \(\Norm_{L/K}(z) \in E(L)\text{.}\) By the previous paragraph, \(E(M) \subseteq E(L)\) whenever \(L \subseteq M\text{,}\) so we may again conclude using the finite intersection property. Alternatively, just note that if each of the (finitely many!) elements of \(E(K)\) fails to survive to some larger field, we can take a compositum to get a single field \(L\) such that no element of \(E(K)\) belongs to \(E(L)\text{,}\) which is absurd since \(E(L) \neq \emptyset\text{.}\)
We now return to the proof of the local existence theorem (Theorem 4.1.5). We first prove the weaker version that does not depend on compatibility of the local reciprocity maps.

Proof.

By the norm limitation theorem (Theorem 4.1.7), it suffices to produce a finite extension \(L/K\text{,}\) not necessarily abelian, such that \(\Norm_{L/K}L^* \subseteq U\text{.}\) This will give us a useful amount of flexibility in what follows.
Let \(m\ZZ \subseteq \ZZ\) be the image of \(U\) in \(K^*/\gotho_K^* \cong \ZZ\text{;}\) by choosing \(L\) to contain the unramified extension of \(K\) of degree \(m\text{,}\) we may ensure that the image of \(\Norm_{L/K} L^*\) in \(K^*/\gotho_K^*\) is also contained in \(m\ZZ\text{.}\) It thus remains to further ensure that
\begin{equation*} (\Norm_{L/K} L^*) \cap \gotho_K^* \subseteq U \cap \gotho_K^*\text{.} \end{equation*}
Since \(\gotho_K^*\) is compact, each open subgroup \((\Norm_{L/K}L^*) \cap \gotho_K^*\) is also closed and hence compact. By Corollary 4.3.16, as \(L/K\) runs over all finite extensions of \(K\text{,}\) the intersection of the groups \((\Norm_{L/K} L^*) \cap \gotho_K^*\) is trivial; in particular, the intersection of the compact subsets
\begin{equation*} ((\Norm_{L/K} L^*) \cap \gotho_K^*) \cap (\gotho_K^* \setminus U) \end{equation*}
of \(\gotho_K^*\) is empty. By the finite intersection property (and taking a compositum), there exists a single \(L/K\) for which \((\Norm_{L/K} L^*) \cap \gotho_K^* \subseteq U \cap \gotho_K^*\text{,}\) as desired.
For the full statement we must use the compatibility of the local reciprocity isomorphisms with subextensions.

Proof.

By Lemma 4.3.17, we can find a finite abelian extension \(M\) of \(K\) such that \(\Norm_{M/K} M^* \subseteq U\text{.}\) By the local reciprocity isomorphism (as obtained in Remark 4.3.7), we then have \(\Gal(M/K) \cong K^*/\Norm_{M/K}M^*\text{.}\) Now take \(L\) to be the fixed field of the subgroup of \(\Gal(M/K)\) corresponding to \(U/\Norm_{M/K}M^*\text{;}\) using Remark 4.3.9, we see that this has the desired effect.

Remark 4.3.19.

We will give an alternate, more explicit construction of abelian extensions of local fields in Section 4.6. However, the approach used in Theorem 4.3.18 is a better analogue of the global setting, where we do not have a convenient explicit construction of abelian extensions to rely on.

Exercises Exercises

1.

Let \(G\) be a finite abelian group and let \(H\) be a subgroup of \(G\text{.}\) Prove that there exists a representation of \(G\) as a product of cyclic groups \(C_1 \times \cdots \times C_m\) of prime power order in such a way that \(H\) is itself equal to \(C'_1 \times \cdots \times C'_m\) for some subgroups \(C'_i\) of \(C_i\text{.}\)

2.

Prove that for any local field \(K\) and any positive integer \(n\) not divisible by the characteristic of \(K\text{,}\) the group \(K^*/(K^*)^{n}\) is finite.
Hint.
See Exercise 1 for a related argument.

3.

Prove that for any local field \(K\) of characteristic \(0\text{,}\) the intersection of the groups \((K^*)^n\) over all positive integers \(n\) is the trivial group.
Hint.
First get the intersection into \(\gotho_K^*\text{,}\) then use prime-to-\(p\) exponents to get it into the 1-units, then use powers of \(p\) to finish. The last step is the only one which fails in characteristic \(p\text{.}\)

4.

Extend Lemma 4.3.13 to the case where \(\ell\) is an arbitrary positive integer, not necessarily prime.
Hint.
It may help to use the structure theorem for finite abelian groups.
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