Skip to main content

Notes on class field theory

Kiran S. Kedlaya

Section 4.3 Local class field theory via Tate’s theorem

Reference.

[36] II.3, III.5.
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 Tate’s theorem

We first prove the theorem of Tate stated earlier (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 do the general case for later use in the global setting.

Proof.

Let \(\gamma\) be a generator of \(H^2(G, M)\text{.}\) Since \(\Cor \circ \Res = [G:H]\) (Example 3.2.22), \(\Res(\gamma)\) generates \(H^2(H,M)\) 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 \(\phi: G^3 \to M\) 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. \end{gather*}
Moreover, \(\phi\) is a coboundary if and only if it’s 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 \(\phi(e, g, hg) = \rho(e,h)^g - \rho(e,hg) + \rho(e,g)\text{.}\)
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 \(H^2(H, M).\)
The map \(\alpha: M[\phi] \to \ZZ[G]\) 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. (Note that this is backwards from the usual exact sequence featuring \(I_G\) as a submodule, which will appear again momentarily.) For any subgroup \(H\) of \(G\text{,}\) we can restrict to \(H\)-modules, then take the long exact sequence:
\begin{equation*} 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). \end{equation*}
To make headway with this, view \(0 \to I_G \to \ZZ[G] \to \ZZ \to 0\) as an exact sequence of \(H\)-modules. Since \(\ZZ[G]\) is induced, its Tate groups all vanish. So we get a dimension shift:
\begin{equation*} H^1(H, I_G) \cong H^0_T(H, \ZZ) = \ZZ/(\#H)\ZZ. \end{equation*}
Similarly, \(H^2(H, I_G) \cong H^1(H, \ZZ) = 0\text{.}\) Also, the map \(H^2(H, M) \to H^2(H, M[\phi])\) is zero because we constructed this map so as to kill off the generator \(\phi\text{.}\) Thus \(H^2(H, M[\phi]) = 0\) and \(H^1(H, I_G) \to H^2(H,M)\) is surjective. But these groups are both finite of the same order! So the map is also injective, and \(H^1(H, M[\phi])\) is also zero.
Now apply Lemma 4.3.2 below 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 conclude that \(H^i_T(G, \ZZ) \cong H^{i+2}_T(G, M)\text{.}\)

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. For \(G\) solvable, we prove the general result by induction on \(\#G\text{.}\) Since \(G\) is solvable, it has a proper subgroup \(H\) for which \(G/H\) is cyclic. By the induction hypothesis, \(H^i_T(H,M) = 0\) for all \(i\text{.}\) Thus by the inflation-restriction exact sequence (Proposition 4.2.14),
\begin{equation*} 0 \to H^i(G/H, M^H) \to H^i(G, M) \to H^i(H, M) \end{equation*}
is exact for all \(i>0\text{.}\) 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 (Theorem 3.4.1), \(H^i_T(G/H, M^H) = 0\) for all \(i\text{,}\) so \(H^i(G/H, M^H) = 0\) for all \(i>0\text{,}\) and \(H^i(G,M) = 0\) for \(i>0\text{.}\) As for \(i=0\text{,}\) note that \(H^0_T(G/H, M^H) = H^2(G/H, M^H) = 0\text{,}\) so for any \(x \in M^G\) there exists \(y \in M^H\) such that \(\Norm_{G/H}(y) = x\text{.}\) Since \(H^0_T(H,M) = 0\text{,}\) there exists \(z \in M\) such that \(\Norm_{H}(z) = x\text{.}\) Now \(\Norm_G(z) = \Norm_{G/H} \circ \Norm_H(z) = x\text{.}\) Thus \(H^0_T(G,M) = 0\text{,}\) as desired.
We next extend the previous argument from \(G\) solvable to \(G\) general (this can be skipped if you only want the final result for solvable \(G\)). For \(i>0\text{,}\) we already know that the group \(H^i(G,M)\) is torsion (Example 3.2.22), so it suffices to show that its \(p\)-primary component vanishes for any prime \(p\text{.}\) To check this, let \(G_p\) be any Sylow \(p\)-subgroup of \(G\text{.}\) As per Example 3.2.22 again, the composition of \(\Res: H^i(G, M) \to H^i(H, M)\) with \(\Cor: H^i(H, M) \to H^i(G, M)\) is multiplication by \([G:G_p]\text{,}\) which is prime to \(p\text{.}\) Consequently, \(\Res\) induces an injective map on \(p\)-primary components. Since \(G_p\) is solvable, we already know that \(H^i(G_p, M) = 0\text{,}\) yielding the desired vanishing. For \(i=0\text{,}\) we argue as in Remark 3.3.13: we know that \(H^0_T(G_p, M) = 0\text{,}\) so the map \(\Norm_{G_p}: M \to M^{G_p}\) is surjective. In particular, for any \(x \in M^G\text{,}\) we can find \(y \in M\) such that \(x = \sum_{g \in G_p} y^g\text{.}\) Then \(\Norm_G(y) = [G:G_p]x\text{,}\) so the group \(H^0_T(G, M)\) is torsion and killed by \([G:G_p]\text{;}\) again varying over \(p\) shows that \(H^0_T(G, M) = 0\text{.}\)
Finally, we treat the case \(i < 0\) by dimension shifting. Make the exact sequence
\begin{equation*} 0 \to N \to \Ind^G_1 M \to M \to 0 \end{equation*}
in which \(m \otimes [g]\) maps to \(m^g\text{.}\) The term in the middle is acyclic, so \(H^{i+1}_T(H', N) \cong H^{i}_T(H', M)\) for any subgroup \(H'\) of \(G\text{.}\) In particular, \(H^1(H', N) = H^2(H', N) = 0\text{,}\) so the above argument gives \(H^i_T(G, N) = 0\) for \(i\geq 0\text{.}\) Now from \(H^0_T(G, N) = 0\) we get \(H^{-1}_T(G, M) = 0\text{;}\) since the same argument applies to \(N\text{,}\) we also get \(H^{-2}_T(G, M) = 0\) and so on.

Subsection Local reciprocity and norm limitation

Let \(L/K\) be a finite Galois extension of local fields. For any intermediate extension \(M/K\text{,}\) we know that \(H^1(L/M) = 0\) and \(H^2(L/M)\) is cyclic of order \([L:M]\text{.}\) We may thus apply Theorem 4.3.1 with \(G = \Gal(L/K)\text{,}\) \(M = L^*\) to obtain isomorphisms \(H^i_T(G, \ZZ) \to H^{i+2}_T(G,M)\text{,}\) thus proving Theorem 4.1.10. This yields a canonical 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{.} \end{equation*}
This establishes the existence of the local reciprocity map (Theorem 4.1.2), keeping in mind that part (a) follows from the explicit computations in Section 4.2), together with the norm limitation theorem (Theorem 4.1.7), modulo one subtlety: if \(M/K\) is another finite Galois extension containing \(L\text{,}\) we need to know that the diagram
Figure 4.3.3.
commutes, so that the maps \(K^* \to \Gal(L/K)^{\ab}\) fit together to give a map \(K^* \to \Gal(K^{\sep}/K)^{\ab}\text{.}\) In other words, we need a commuting diagram
Figure 4.3.4.
The right square in Figure 4.3.4 comes from taking long exact sequences in the commutative diagram with exact rows:
Figure 4.3.5.
To construct the left square in Figure 4.3.4, we similarly need to construct a commutative diagram with exact rows:
Figure 4.3.6.
I claim we can arrange for this as follows. First choose a cocycle \(\phi_M: \Gal(M/K)^3 \to M^*\) representing the preferred generator of \(H^2(M/K)\text{.}\) Then there exists a unique map \(\phi_L\) fitting into the following commuting square:
Figure 4.3.7.
and this will necessarily give a cocycle representing the preferred generator of \(H^2(L/K)\text{.}\) Further details omitted.

Subsection The local existence theorem

It remains to prove 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 1), 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 law (Theorem 4.1.2), \(K^*/\Norm_{M/K}M^* \cong (\ZZ/\ell \ZZ)^n\text{;}\) 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 4.3.9.

The conclusion of Lemma 4.3.8 remains true even if \(\ell\) is not prime; see Exercise 3. This statement can be interpreted in terms of the Hilbert symbol, whose properties generalize quadratic reciprocity to higher powers; see [36], III.4.
This allows to deduce a corollary of the existence theorem which is needed in its proof. (The argument we give here depends squarely on characteristic \(0\text{;}\) some patching is needed in the positive characteristic case.)

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 4.3.8, every element of \(D_K\) is an \(\ell\)-th power in \(K\) for every prime \(\ell\text{.}\) 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 \(n\text{.}\) This will then imply using Exercise 2 that \(D_K\) is the trivial group. (Alternatively, one can follow the suggestion of Remark 4.3.9 and prove that the conclusion of Lemma 4.3.8 retains true when \(\ell\) is replaced by an arbitrary positive integer \(n\text{,}\) and then apply Exercise 2 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).

Proof.

We note first that by the local reciprocity law (Theorem 4.1.2), it is enough to construct \(L\) so that \(U\) contains \(\Norm_{L/K}L^*\text{:}\) in this case, we will have \(\Gal(L/K) \cong K^*/\Norm_{L/K}L^*\text{,}\) and then \(U/\Norm_{L/K}L^*\) will corresponding to \(\Gal(L/M)\) for some intermediate extension \(M/K\) having the desired effect. We note next that by the norm limitation theorem (Theorem 4.1.7), it suffices to produce any finite extension \(L/K\text{,}\) not necessarily abelian, such that \(U\) contains \(\Norm_{L/K}L^*\text{.}\)
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^*. \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 4.3.10, 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.

Exercises Exercises

1.

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.

2.

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 \(p\text{.}\)

3.

Extend Lemma 4.3.8 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.
PrevTopNext