Sets of Completely Decomposed Primes in Extensions of Number Fields

by Kay Wingberg

June 15, 2014

Let p be a prime number and let k(p) be the maximal p-extension of a number field k. If T is a set of primes of k, then k^{T} (p) denotes the maximal p-extension of k which is completely decomposed at T. Assuming that T is finite, the canonical homomorphism

ϕ^{T} (p) : ∗_{𝔭∈T (k^{T} (p))}G_{𝔓}(k(p)|k) ⟶ G(k(p)|k^{T} (p))

of the free pro-p-product of the decomposition groups G_{𝔓}(k(p)|k) into the Galois group G(k(p)|k^{T} (p)) is an isomorphism, see [3] theorem (10.5.8); here the prime 𝔓 is an arbitrary extension of 𝔭 to k(p).

In the profinite case, i.e. considering the maximal Galois extension k^{T} which is completely decomposed at the finite set T, there exist suitable extensions 𝔓|𝔭 such that ϕ^{T} is an isomorphism of profinite groups. If T = S_{∞} is the set archimedean primes, this is a result of Fried-Haran-Völklein [1] and in general it is proven by Pop [4].

The fact that ϕ^{T} (p) is an isomorphism if T is finite implies very strong properties for the extension k^{T} (p)|k. In particular, the (strict) cohomological dimension of the Galois group G(k^{T} (p)|k) is equal to 2 (if p = 2 one has to require that k is totally imaginary). Furthermore, we get for the corresponding local extensions that

(∗)  (k^{T} (p))_{𝔓} = k_{𝔭}(p) for all primes 𝔓|𝔭, 𝔭 ∉ T,

i.e. k^{T} (p) realizes the maximal p-extension k_{𝔭}(p) of the local fields k_{𝔭} for all primes not in T. In particular, the set T is equal to the set D(k^{T} (p)|k) of all primes of k which decomposed completely in the extension k^{T} (p)|k. We will say that T is saturated if it has this property and we call the stronger property (∗) that T is strongly saturated. If the Dirichlet density δ(T) is positive, then T is saturated if and only if T is the set of completely decomposed primes of a finite Galois extension of k.

If T is an arbitrary set of primes of k, we call the set T̂ = D(k^{T} (p)|k) the saturation of T. The most interesting case is that T is an infinite set of primes of density zero. If T is saturated, then the extension k^{T} (p)|k is infinite. We will see that there exist infinite sets T of primes such that δ(T̂) > δ(T) = 0, and also sets T such that δ(T̂) = δ(T) = 0. An important example is the following.

Theorem 1: Let p be an odd prime number and let k be a CM-field containing the group μ_{p} of all p-th roots of unity, with maximal totally real subfield k^{+}, i.e. k = k^{+}(μ_{p}) is totally imaginary and [k : k^{+}] = 2. Let S_{p} = {𝔭|p} and

T = {𝔭 | 𝔭 ∩ k^{+} is inert in k|k^{+}}.

Then T ∪ S_{p} is strongly saturated.

Moreover, in the example above we get that the Galois group

G((k^{T})_{nr}(p)|k^{T} (p))

of the maximal unramified p-extension of (k^{T})_{nr}(p) of k^{T} (p) is a free pro-p-group. This will follow from a more general theorem which deals with a generalization of the notion of saturated sets.

A set T = T (k) is called stably saturated if the sets T (k′) are saturated for every finite Galois extension k′|k inside k^{T} (p). These sets are necessarily of density 0 if they are not equal to set P of all primes. Obviously, strongly staturated sets are stably saturated. We have the following theorem.

Theorem 2: Let T ≠ P be a stably saturated set of primes of a number field k. Then the canonical map

ϕ^{T} (p) : ∗_{𝔓∈T (k^{T} (p))}(G_{𝔓}(k(p)|k), T_{𝔓}(k(p)|k)) ⟶^{∼} G(k(p)/k^{T} (p))

is an isomorphism.

Here ∗_{𝔓∈T (k^{T} (p))}(G_{𝔓}(k(p)|k), T_{𝔓}(k(p)|k)) denotes the free corestricted prop-product of the decomposition groups G_{𝔓}(k(p)|k) with respect to the inertia groups T_{𝔓}(k(p)|k), see [2].

Corollary: In the situation of theorem 1 let T̃ = T ∪ S_{p}. Then

∗_{𝔓∈T̃(kT̃(p))}(G_{𝔓}(k(p)|k), T_{𝔓}(k(p)|k)) ⟶^{∼} G(k(p)/kT̃(p)).

Finally I would like to thank Jochen Gärtner for helpful discussions on this subject and Alexander Ivanov for valuable comments.

1 Saturated sets of primes of a number field

We start with some remarks on complete lattices. Let

\xymatrix{A \ar[r]^{φ} & B \ar[l]^{ψ}}

be maps between complete lattices (A, ⊆) and (B, ⊆) with the following properties:

I. φ and ψ are order-reversing, II. A ⊆ ψφ(A) and B ⊆ φψ(B) for all A ∈ A and B ∈ B.

For A ∈ A and B ∈ B we define the saturation

Â := ψφ(A) and B̂ := φψ(B),

and call A ∈ A resp. B ∈ B to be saturated if A = Â resp. B = B̂. We put

A_{sat} = {A ∈ A | A is saturated}, B_{sat} = {B ∈ B | B is saturated},

and we have the following properties:

(i) A_{1} ⊆ A_{2} implies Â_{1} ⊆ Â_{2} and B_{1} ⊆ B_{2} implies B̂_{1} ⊆ B̂_{2}. (ii) φψφ = φ, ψφψ = ψ, ˆ Â = Â, ˆ B̂ = B̂.

(iii) B_{sat} is the image of A under φ and A_{sat} is the image of B under ψ, i.e. φ : A ↠ B_{sat} and ψ : B ↠ A_{sat}, and ψ and φ induce bijections

\xymatrix{A_{sat} \ar[r]^{φ} & B_{sat}. \ar[l]^{ψ}}

(iv)

φ(⋃_{i} A_{i}) = ⋂_{i} φ(A_{i}), φ(⋂_{i} ψ(B_{i})) = { \atop ⋃_{i} B_{i}}.

In particular, the infimum of saturated elements is again saturated. The verification of these statements is straightforward using the properties I and II of the maps φ and ψ. We define the equivalence relations

A_{1} ∼ A_{2} :⇔ φ(A_{1}) = φ(A_{2}) for A_{1}, A_{2} ∈ A

and

B_{1} ∼ B_{2} :⇔ ψ(B_{1}) = ψ(B_{2}) for B_{1}, B_{2} ∈ B,

and denote the classes by [X]. Obviously, X ∼ X̂ by (ii) and for Y ∈ [X] we have Y ⊆ X̂, and so X̂ is the unique maximal element of [X]. Furthermore, the surjections ψφ : A ↠ A_{sat} and φψ : B ↠ B_{sat} induce bijections

ψφ : (A/_{∼}) ⟶^{∼} A_{sat}, [A] ↦ Â, and φψ : (B/_{∼}) ⟶^{∼} B_{sat}, [B] ↦ B̂.

Now let K be a number field. We use the following notation: S_{∞}, S_{ℝ} and S_{ℂ} are the sets of archimedean, real and complex primes of K, respectively, P is the set of all primes of K, and if p is a prime number, then S_{p} is the set of all primes of K above p.

If 𝔭 is a prime of K, then K_{𝔭} is the completion of K with respect to 𝔭. If L|K is a Galois extension, then we denote the decomposition group and inertia group of the Galois group G(L|K) with respect to 𝔭 by G_{𝔭}(L|K) and T_{𝔭}(L|K), respectively.

For a set S of primes of K, let δ(S) = δ_{K}(S) be its Dirichlet density. If S = S(K) is a set of primes and K′|K an algebraic extension of K, then we denote the set of primes of K′ consisting of all prolongations of S by S(K′).

If 𝔠 is a class of finite groups which is closed under taking subgroups, homomorphic images and group extensions, then K(𝔠) is the maximal pro-𝔠-extension of K, and in particular if p is a prime number, K(p) denotes the maximal pextension of K. By abuse of notation, we denote the maximal pro-𝔠 extension of K which is completely decomposed at T by K^{T} (𝔠), and K^{T} (p) is the maximal p-extension of K inside K^{T}.

Let

\xymatrix{E_{K} =\{L | L is a Galois extension of K\} \ar[r]^{φ} & \{T | T is a set of primes of K\}=S_{K} \ar[l]^{ψ}}

where

φ(L) = D(L|K) is the set of primes which are completely decomposed in L|K,
ψ(T) = K^{T}  is the maximal Galois extension of K which is completely

decomposed at T .

Obviously, the maps φ and ψ are order-reversing and

L ⊆ ψφ(L) = K^{D(L|K)} =: L̂ and T ⊆ φψ(T) = D(K^{T} |K) =: T̂.

Furthermore, we define the equivalence relations

L_{1} ∼ L_{2} :⇔ D(L_{1}|K) = D(L_{2}|K) for L_{1}, L_{2} ∈ E_{K}

and

T_{1} ∼ T_{2} :⇔ K^{T_{1}} = K^{T_{2}} for T_{1}, T_{2} ∈ S_{K}.

Definition 1.1 The extension L ∈ E_{K} is called saturated if L̂ = L, i.e.

L = K^{D(L|K)},

and the set T ∈ S_{K} is called saturated if T̂ = T, i.e.

T = D(K^{T} |K).

We strengthen the notion of saturated sets in the following way:

Definition 1.2 Let T be a set of primes of K.

(i) The set T = T (K) is called stably saturated if the sets T (K′) ∈ S_{K′} are saturated for every finite Galois extension K′|K inside K^{T}.

(ii) We call T to be strongly saturated if

(K^{T})_{𝔓} = K̄_{𝔭} for all primes 𝔓|𝔭, 𝔭 ∉ T,

where K̄_{𝔭} is the algebraic closure of K_{𝔭}.

Remark 1: If we consider the set E_{K}(𝔠) = {L | L is a pro-𝔠-extension of K}, we define a set T ∈ S_{K} to be 𝔠-saturated if T = D(K^{T} (𝔠)|K), and an extension L ∈ E_{K}(𝔠) is called 𝔠-saturated if L = K^{D(L|K)}(𝔠). Analogously, we define stably 𝔠-saturated and strongly 𝔠-saturated, e.g. T is strongly 𝔠-saturated if (K^{T} (𝔠))_{𝔓} = K_{𝔭}(𝔠) for all primes 𝔓|𝔭, 𝔭 ∉ T.

We say, a prime of K is redundant (or more precise, 𝔠-redundant) if is totally decomposed in every extension inside K(𝔠). Obviously, redundant primes are necessarily archimedean primes and the complex primes are always redundant. But also real primes might be redundant if we restrict to pro-𝔠-extensions, e.g. if we consider p-extensions where p is an odd prime number. Therefore we make the following

Convention: In the following all considered primes are not redundant and S_{K} = S_{K}(𝔠) consists only of sets of non-redundant primes for the extension K(𝔠)|K.

Immediately from the definitions (and the convention) above we get

Lemma 1.3 Let T ∈ S_{K}.

(i) T is saturated if and only if (K^{T})_{𝔭} ≠ K_{𝔭} for all 𝔭 ∉ T.

(ii) T is stably saturated if and only if (K^{T})_{𝔭}|K_{𝔭} is an infinite extension for all 𝔭 ∉ T.

From the general remarks on partial ordered sets we see that there are bijections

\xymatrix{(E_{K})_{sat} \ar[r]^{φ} & (S_{K})_{sat} \ar[l]^{ψ}}

and we have the following Lemma 1.4

(i) T_{1} ⊆ T_{2} implies T̂_{1} ⊆ T̂_{2} and L_{1} ⊆ L_{2} implies L̂_{1} ⊆ L̂_{2}.

(ii) K^{T} = KT̂ and D(L|K) = D(L̂|K).

(iii) D(∏_{i} L_{i}|K) = ⋂_{i} D(L_{i}|K) and K^{⋃_{i} T_{i}} = ⋂_{i} K^{T_{i}}.

(iv) D(⋂_{i} K^{T_{i}}|K) = ; ⋃_{i} T_{i} and K^{⋂_{i} D(L_{i}|K)} = ; ∏_{i} L_{i}.

Theorem 1.5

(i) If T is a finite set of primes of K, then T is strongly saturated.

(ii) If L is a finite Galois extension of K, then L is saturated.

Proof: (i) Let p be a prime number and let L|K be a finite Galois extension inside K^{T}. Let 𝔭_{0} ∉ T, 𝔓_{0} a fixed extension of 𝔭_{0} to K^{T} and \overline{𝔓}_{0} the restriction of 𝔓_{0} to L. By the theorem of Grunwald/Wang (see [3], theorem (9.2.2)) the canonical homomorphism

H^{1}(L, ℤ/pℤ) ⟶ H^{1}(L_{\overline{𝔓}_{0}}, ℤ/pℤ) ⊕ ⨁_{𝔓∈T (L)}H^{1}(L_{𝔓}, ℤ/pℤ)

is surjective. In particular, for every α_{\overline{𝔓}_{0}} ∈ H^{1}(L_{\overline{𝔓}_{0}}, ℤ/pℤ)) there exists an element β ∈ H^{1}(L, ℤ/pℤ) which is mapped to (α_{\overline{𝔓}_{0}}, 0, …, 0). But β lies in the subgroup H^{1}(K^{T} |L, ℤ/pℤ) of H^{1}(L, ℤ/pℤ). Therefore

H^{1}(K^{T} |L, ℤ/pℤ) ⟶ H^{1}(L_{\overline{𝔓}_{0}}, ℤ/pℤ)

is surjective. Varying L and p, it follows that the completion of K^{T} with respect to the prime 𝔓_{0}, 𝔓_{0}|𝔭_{0} and 𝔭_{0} ∉ T, is equal to the algebraic closure of K_{𝔭_{0}} (since G(\overline{K}_{𝔭_{0}}|K_{𝔭_{0}}) is pro-solvable).

(ii) Let L′ be a finite Galois extension of K with L ⊆ L′ ⊆ K^{D(L|K)}. Since D(L|K) ⊆ D(L′|K), we obtain for the densities of these sets the inequality δ(D(L|K)) ≤ δ(D(L′|K)), and so, by Čebotarev’s density theorem,

[L′ : K] = δ(D(L′|K))^{−1} ≤ δ(D(L|K))^{−1} = [L : K].

This shows that L′ = L and so L = K^{D(L|K)}. □

Now we consider sets of primes which are infinite and of density equal to 0.

Proposition 1.6 For a set T of primes we have

δ(T̂) = 0 ⇔ K^{T} |K is an infinite extension.

Proof: Since T̂ = D(K^{T} |K), it follows that δ(T̂) > 0 if K^{T} |K is a finite extension. Conversely, assume that K^{T} |K is infinite and let L be a finite Galois extension of K inside K^{T}. Then

\frac{1}{[L : K]} = δ(D(L|K)) ≥ δ(T̂),

and so δ(T̂) = 0. □

Remark 2: From the proposition above and lemma (1.3)(ii) it follows that a stably saturated set of primes T ≠ P has necessarily density equal to 0.

Remark 3: Let n ∈ ℕ. Then there exist infinite sets T of primes such that

\frac{1}{n} = δ(T̂) > δ(T) = 0.

Example 1: Let L|K be a Galois extension of the number field K such that [L : K] = n. The set E_{L|K}^{fin} of proper finite extensions of L being Galois over K is countable, say E_{L|K}^{fin} = {L_{i}, i ∈ ℕ}. We choose for every i ∈ ℕ an element σ_{i} ∈ G(L_{i}|L), σ_{i} ≠ 1, and a prime 𝔭_{i} of K which is unramified in L_{i}|K having a Frobenius (\frac{L_{i}|K}{𝔓_{i}}) = σ_{i}, 𝔓_{i}|𝔭_{i}. Since there exist infinitely many such primes, there is a 𝔭_{i} such that N_{K|ℚ}𝔭_{i} ≥ i^{2}. Then

∑_{i=1}^{∞} \frac{1}{N_{K|ℚ}𝔭_{i}} ≤ ∑_{i=1}^{∞} \frac{1}{i^{2}}

converges, and so the set T = {𝔭_{i}, i ∈ ℕ} has density equal to 0. Furthermore we have

K^{T} = L.

Indeed, since every 𝔭 ∈ T is completely decomposed in L, we have L ⊆ K^{T} and a finite Galois extension E|K, L ⊊ E ⊆ K^{T} would be a field L_{i} for some i ∈ ℕ and so 𝔭_{i} ∈ T would not be completely decomposed in E|K. Therefore T̂ = D(L|K) and δ(T̂) = \frac{1}{n} > 0.

Remark 4: There exist infinite sets T of primes such that

δ(T̂) = δ(T) = 0.

Example 2: Assume that K is a number field such that

(i) K is not totally real, (ii) there exists a proper subfield E of K such that K|E is a cyclic extension.

We denote the Galois group G(K|E) by Δ. Let

T_{0} = {𝔭 | 𝔭 ∩ E is inert in K|E}.

Then T_{0} is an infinite set of primes of K of density equal to 0. Let p be a prime number not dividing [K : E] whose extensions to E are completely decomposed or totally ramified in K|E. Let K_{∞} resp. E_{∞} be the compositum of all ℤ_{p}-extension of K resp. E. Then

G(K_{∞}|K) = ℤ_{p}^{r_{2}(K)+1+δ_{K}}, G(E_{∞}|E) = ℤ_{p}^{r_{2}(E)+1+δ_{E}},

where r_{2} denotes the number of complex places and δ is the so-called Leopoldt defect. We have a decomposition of ℤ_{p}[Δ]-modules

G(K_{∞}|K) ≅ G(E_{∞}|E) ⊕ M,

where M is a Δ-module with M^{Δ} = 0 and r = rang_{ℤ_{p}}M ≥ r_{2}(K) − r_{2}(E) > 0, since δ_{K} ≥ δ_{E}. Let L be the subfield of K_{∞} corresponding to G(K_{∞}|K)^{Δ} ≅ G(E_{∞}|E), i.e.

G(L|K) ≅ M

and so G(L|K)^{Δ} = 0. Observe that L is a Galois extension of E. Now let 𝔭 ∈ T_{0}. For the decomposition group with respect to 𝔭 we have the split exact sequence

0 ⟶ G_{𝔭}(L|K) ⟶ G_{𝔭}(L|E) ⟶ Δ_{𝔭} ⟶ 0.

If G_{𝔭}(L|K) would be non-trivial, then G_{𝔭}(L|E) is not abelian since Δ_{𝔭} = Δ ≠ 0 acts non-trivially on G_{𝔭}(L|K) ⊆ G(L|K). But 𝔭 lies not above p and so it is unramified in L|K, thus unramified in L|E and so G_{𝔭}(L|E) is cyclic. Therefore all primes in T_{0} are completely decomposed in the infinite extension L|K. Let

T = D(L|K).

Then T is an infinite saturated set (containing T_{0}), i.e. T = T̂, and since K^{T} is an infinite extension (containing L), it follows from proposition (1.6) that δ(T̂) = 0.

Furthermore, if S_{ℝ} ⊆ T, then

(K^{T})_{𝔭}|K_{𝔭} is an infinite extension if 𝔭 ∉ T ,

hence T is stably saturated. This follows from the fact, that for a prime 𝔭 ∉ D(L|K) the non-trivial decomposition group G_{𝔭}(L|K) has to be a infinite subgroup of G(L|K) ≅ ℤ_{p}^{r}.

An example for the situation above is a CM-field K with maximal totally real subfield K^{+}. If T_{0}(K^{+}) = {𝔭 | 𝔭 is inert in K}, then T_{0} = T_{0}(K) is infinite of density equal to zero. Let K_{ac} be the anti-cyclotomic ℤ_{p}-extension of K, p ≠ 2, and T = D(K_{ac}|K). Then T_{0} ⊆ T = T̂ and so K^{T} |K is an infinite extension and therefore δ(T̂) = 0.

Now we consider the cardinality of an equivalence class [T] of a set of primes T of K. By definition, the saturation T̂ of T is the unique maximal element of [T] (with respect to the inclusion). But the example 1 of remark 3 shows that there might be infinitely many different (even pairwise disjoint) minimal elements in the class [T].

Proposition 1.7

(i) If L is a finite Galois extension of K, then

#[L] = 1 and #[D(L|K)] = ∞.

(ii) If T is a finite set of primes, then

#[T] = 1 and #[K^{T}] = ∞.

Proof: Let L′ ∈ [L]. Then L′ ⊆ L̂ = L by theorem (1.5)(ii). Thus L′|K is finite and so

L′ = K^{D(L′|K)} = K^{D(L|K)} = L.

In order to prove the second assertion of (i), let L be a finite Galois extension of K and let T ⊆ D(L|K) be a subset of density equal to zero. Then

L = K^{D(L|K)} = K^{D(L|K)\T} ,

hence every subset D(L|K)\T with δ(T) = 0 is contained in [D(L|K)].

By theorem (1.5)(i), every subset of T is (strongly) saturated. Thus the first assertion of (ii) follows. For the second let S = T ∪{𝔭}, where 𝔭 is some prime not in T. By [3](10.5.8) we have for the Galois group of the extension K^{T} (p)|K^{S}(p) the isomorphism

∗_{𝔭∈S\T (K^{S}(p))}G(K_{𝔭}(p)|K_{𝔭}) ⟶^{∼} G(K^{T} (p)|K^{S}(p)),

where p is some prime number and E(p)|K denotes the maximal p-extension inside a Galois extension E|K. In particular, the extension K^{T} |K^{S} is infinite. Since every Galois extension L|K with K^{S} ⊊ L ⊆ K^{T} is contained in [K^{T}], the second assertion of (ii) follows. □

2 The maximal p-extension k^{T}(p) of k

In the following we will consider p-extensions and by a saturated set of primes we always mean a p-saturated set (see remark 1 of the first section).

Let S, T be sets of primes of a (not necessarily finite) number field K and let 𝔭 be a prime of K. Let K(p) be the maximal p-extension of K. Mostly we will drop the notion (p). So let

K_{nr} is the maximal unramified p-extension of K,

K_{S} is the maximal p-extension of K which is unramified outside S,

K^{T} is the maximal p-extension of K which is completely decomposed at T,

K_{S}^{T} is the maximal p-extension of K which is unramified outside S and

completely decomposed at T,

G_{𝔭}(K) = G_{𝔭}(K(p)|K) ≅ G(K_{𝔭}(p)|K_{𝔭}) is the decomposition group,

T_{𝔭}(K) = T_{𝔭}(K(p)|K) ≅ T (K_{𝔭}(p)|K_{𝔭}) is the inertia group with respect to 𝔭.

In the following k will always denote a finite number field. If K|k is an infinite extension of number fields and S a set of primes of k, then S(K) denotes the profinite space

S(K) = lim_{⟵}_{k′} S(k′) ∪ {∗_{k′}}

where k′ runs through the finite subextensions of K|k and S(k′) ∪ {∗_{k′}} is the one-point compactification of the discrete set S(k′) of primes of k′ lying above S. According to [3] (10.5.8) and (10.5.10) we have

Theorem 2.1 If R′ ⊆ R ⊆ S ⊆ S′ are sets of primes of k such that δ(S) = 1 and R is finite, then the canonical homomorphism

ϕ_{S′,S}^{R′,R} : ∗_{𝔭∈S′\S(k_{S}^{R})}T (k_{𝔭}(p)|k_{𝔭}) ∗ ∗_{𝔭∈R\R′(k_{S}^{R})}G(k_{𝔭}(p)|k_{𝔭}) ⟶ G(k_{S′}^{R′}|k_{S}^{R})

is an isomorphism. Furthermore we have the following assertions concerning the (strict) cohomological dimension: Assume that k is totally imaginary if p = 2, then

cd_{p} G(k_{S}^{R}|k) = scd_{p} G(k_{S}^{R}|k) = 2.

If L|K is a Galois extension we write H^{i}(L|K, A) for H^{i}(G(L|K), A), and for a pro-p group G we put H^{i}(G) = H^{i}(G, ℤ/pℤ). For a p-extension K|k we will use the notation

\begin{matrix}⨁ & ′ \\ 𝔭∈T (K) \end{matrix}H^{i}(G_{𝔭}(k(p)|K)) := \underrightarrow{lim}_{k′} ⨁_{𝔭∈T (k′)}H^{i}(G_{𝔭}(k(p)|k′)

where k′ runs through all finite subextensions of K|k.

Proposition 2.2 Let p be a prime number and let T be a set of primes of a number field k of density δ(T) = 0. Then the following assertions are equivalent:

(i) T is stably saturated. (ii) cd_{p} G_{𝔭}(k(p)|k^{T}) ≤ 1 for all 𝔭 ∉ T. (iii) The canonical map

H^{2}(k(p)|k^{T}) ⟶^{∼} \begin{matrix}⨁ & ′ \\ 𝔭∈T (k^{T}) \end{matrix}H^{2}(G_{𝔭}(k))

is an isomorphism. (iv) The group G((k^{T})_{S}|k^{T}) is free for every S with δ(S) = 1, S ∩ T = ∅.

Proof: (i)⇔(ii): By lemma (1.3) the set T is stably saturated if and only if the extensions (k^{T})_{𝔭}|k_{𝔭} are infinite for all primes 𝔭 ∉ T, i.e. if and only if cd_{p} G_{𝔭}(k(p)|k^{T}) ≤ 1, see [3] (7.1.8)(i).

In order to prove (ii)⇔(iii), first observe that k^{T} |k is an infinite extension as δ(T) = 0. Using the Poitou-Tate theorem, see [3] (8.6.10), (10.4.8), and the Hasse principle, loc. cit. (9.1.16), and passing to the limit over all finite extensions k′ inside k^{T} |k, we obtain the isomorphism

H^{2}(k(p)|k^{T}) ⟶^{∼} \begin{matrix}⨁ & ′ \\ 𝔭∈T (k^{T}) \end{matrix}H^{2}(G_{𝔭}(k)) ⊕ \begin{matrix}⨁ & ′ \\ 𝔭∉T (k^{T}) \end{matrix}H^{2}(G_{𝔭}(k(p)|k^{T})),

since \underrightarrow{lim}_{k′} H^{0}(G(k(p)|k′), μ_{p})^{∨} = 0. This shows that (iii) is equivalent to

H^{2}(G_{𝔭}(k(p)|k^{T})) = 0 for all 𝔭 ∉ T (k^{T}),

and so we get (ii)⇔(iii).

From the Poitou-Tate exaxt sequences and the Hasse principle for the extensions k(p)|k and k_{S}|k (using δ(S) = 1) we get the exact sequence

0 ⟶ H^{2}(k_{S}|k) ⟶ H^{2}(k(p)|k) ⟶ ⨁_{𝔭∉S}H^{2}(G_{𝔭}(k)) ⟶ 0.

Passing to the limit from k to k^{T}, we obtain the exact sequence

0 → H^{2}((k^{T})_{S}|k^{T}) → H^{2}(k(p)|k^{T}) → \begin{matrix}⨁ & ′ \\ 𝔭∉S(k^{T}) \end{matrix}H^{2}(G_{𝔭}(k^{T})) → 0.

Assuming (ii), we have H^{2}(G_{𝔭}(k(p)|k^{T})) = 0 for all 𝔭 ∉ T (k^{T}), and so

\begin{matrix}⨁ & ′ \\ 𝔭∉S(k^{T}) \end{matrix}H^{2}(G_{𝔭}(k(p)|k^{T})) = \begin{matrix}⨁ & ′ \\ 𝔭∈T (k^{T}) \end{matrix}H^{2}(G_{𝔭}(k(p)|k^{T})) = \begin{matrix}⨁ & ′ \\ 𝔭∈T (k^{T}) \end{matrix}H^{2}(G_{𝔭}(k))

for every S with S ∩ T = ∅. It follows that H^{2}((k^{T})_{S}|k^{T}) = 0, i.e. assertion (iv) holds.

Finally, if H^{2}((k^{T})T̄|k^{T}) = H^{2}(kT̄|k^{T}) = 0 for T̄ = P\T, then, using again the exact sequence above, assertion (iii) holds. □

There is another situation in which the pro-p group G((k^{T})_{S}|k^{T}) is free.

Theorem 2.3 Let p be a prime number and let T and S are sets of primes of a number field k, where T ∪ S ≠ P is strongly saturated and T ∩ S = ∅. Then the following holds:

(i) (k^{T})_{S} = kT̄ and cd_{p} G((k^{T})_{S}|k^{T}) ≤ 1.

(ii) If δ(S) = 0, then cd_{p} G((k^{T})_{nr}|k^{T}) ≤ 1, i.e. the Galois group of the maximal unramified p-extension (k^{T})_{nr}|k^{T} is a free pro-p-group.

In particular, if T ≠ P is a strongly saturated set, then

cd_{p} G((k^{T})_{nr}|k^{T}) ≤ 1.

Proof: Since T ∪ S ≠ P is stably saturated, we have δ(T) ≤ δ(T ∪ S) = 0, hence δ(T̄) = 1, where T̄ = P\T. It follows from proposition (2.2) that the group G(kT̄|k^{T}) is free. Since T ∪ S is strongly saturated, the extension k^{T ∪S} realizes the local extensions for all primes in \overline{T ∪ S} = T̄\S, and so the extension k^{T} has this property. Therefore kT̄|k^{T} is completely decomposed by \overline{T ∪ S}, hence kT̄ = (k^{T})T̄ = (k^{T})_{S}. This proves (i).

Now assume that δ(S) = 0, hence δ(T̄\S) = 1. Let K|k be a finite extension inside k^{T}. Using the Poitou-Tate theorem and the Hasse principle, we see that the canonical map

H^{2}(K_{T̄\S}|K) ⟶ H^{2}(kT̄|K)

is injective. Passing to the limit, it follows that

H^{2}((k^{T})_{T̄\S}|k^{T}) ⟶ H^{2}(kT̄|k^{T})

is injective. Since (k^{T})_{T̄\S} = (k^{T})_{nr}, the desired result follows from (i). □

By theorem (1.5)(i) finite sets are strongly saturated. Now we will show that there are also infinite strongly saturated sets.

Theorem 2.4 Let p be an odd prime number and let k be a CM-field containing the group μ_{p} of all p-th roots of unity, with maximal totally real subfield k^{+}, i.e. k = k^{+}(μ_{p}) is totally imaginary and [k : k^{+}] = 2. Let

T = {𝔭 | 𝔭 ∩ k^{+} is inert in k|k^{+}}.

Then the set T_{0} = T ∪ S_{p} of primes of k is strongly saturated. Furthermore, the Galois group G((k^{T_{0}})_{nr}|k^{T_{0}}) is a free pro-p-group.

Proof: Let

S = {𝔭 | 𝔭 ∩ k^{+} is decomposed in k|k^{+}}

and S_{1} = P\T_{0} = S\S_{p}. Let S_{0} be a subset of S_{1} invariant under the action of G(k|k^{+}) such that V = S_{1}\S_{0} is finite (observe that T (k_{𝔭}(p)|k_{𝔭}) is cyclic for 𝔭 ∈ V as V ∩ S_{p} = ∅). Let K|k be a finite extension inside k^{T_{0}} being Galois over k^{+} (observe that k^{T_{0}}|k^{+} is a Galois extension as T_{0} is invariant under G(k|k^{+})). First we show the following

Claim: There exists an abelian (not necessarily finite) p-extension L|K, which is Galois over k^{+}, central over k, unramified by T_{0}(K), completely decomposed at S_{p} and ramified at each prime of V (K):

T (k_{𝔭}(p)|K_{𝔭})_{G_{𝔭}(k(p)|k)} ⊆ G_{𝔭}(L|K) for all 𝔭 ∈ V (k).

Proof : Consider the group extension

1 ⟶ G(K_{S_{1}∪S_{p}}^{S_{p}}|K_{S_{0}∪S_{p}}^{S_{p}}) ⟶ G(K_{S_{1}∪S_{p}}^{S_{p}}|K) ⟶ G(K_{S_{0}∪S_{p}}^{S_{p}}|K) ⟶ 1.

Since δ(S_{0}) = 1, we have H^{2}(G(K_{S_{0}∪S_{p}}^{S_{p}}|K), ℚ_{p}/ℤ_{p}) = 0, see (2.1). In the proof of this claim we write H^{i}(E|F) for H^{i}(G(E|F), ℚ_{p}/ℤ_{p}). We obtain an exact sequence

0 → H^{1}(K_{S_{0}∪S_{p}}^{S_{p}}|K) → H^{1}(K_{S_{1}∪S_{p}}^{S_{p}}|K) → H^{1}(K_{S_{1}∪S_{p}}^{S_{p}}|K_{S_{0}∪S_{p}}^{S_{p}})^{G(K_{S_{0}∪Sp}^{Sp}|K)} → 0,

and so an exact sequence

0 → H^{1}(K_{S_{0}∪S_{p}}^{S_{p}}|K)^{G(K|k)} → H^{1}(K_{S_{1}∪S_{p}}^{S_{p}}|K)^{G(K|k)} → H^{1}(K_{S_{1}∪S_{p}}^{S_{p}}|K_{S_{0}∪S_{p}}^{S_{p}})^{G(K_{S_{0}∪Sp}^{Sp}|k)}

→ H^{1}(K|k, H^{1}(K_{S_{0}∪S_{p}}^{S_{p}}|K)) → H^{1}(K|k, H^{1}(K_{S_{1}∪S_{p}}^{S_{p}}|K)).

Using again that H^{2}(G(KS_{p} ; S̃|K), ℚ_{p}/ℤ_{p}) = 0 where S̃ = S_{0} ∪S_{p} resp. S̃ = S_{1} ∪S_{p}, the Hochschild-Serre spectral sequences

E_{2}^{i,j} = H^{i}(K|k, H^{j}(KS_{p} ; S̃|K)) ⇒ H^{i+j}(KS_{p} ; S̃|k)

show that in the commutative diagram

[figure: H^{1}(K|k, H^{1}(K_{S_{0}∪S_{p}}^{S_{p}}|K)); d_{2}^{2,1}; H^{1}(K|k, H^{1}(K_{S_{1}∪S_{p}}^{S_{p}}|K)); d_{2}^{2,1}; H^{3}(K|k, H^{0}(K_{S_{0}∪S_{p}}^{S_{p}}|K)); H^{3}(K|k, H^{0}(K_{S_{1}∪S_{p}}^{S_{p}}|K))]

the differentials d_{2}^{2,1} are isomorphisms. Thus we obtain an exact sequence

0 ⟶ (G(K_{S_{1}∪S_{p}}^{S_{p}}|K_{S_{0}∪S_{p}}^{S_{p}})^{ab})_{G(K_{S_{0}∪Sp}^{Sp}|k)} ⟶

(G(K_{S_{1}∪S_{p}}^{S_{p}}|K)^{ab})_{G(K|k)} ⟶ (G(K_{S_{0}∪S_{p}}^{S_{p}}|K)^{ab})_{G(K|k)} ⟶ 0.

Using (2.1), we obtain the isomorphism

∏_{𝔭∈V (k)}T (k_{𝔭}(p)|K_{𝔭})_{G_{𝔭}(k(p)|k)} _{∼} →^{can} G(K_{S_{1}∪S_{p}}^{S_{p}}|K_{S_{0}∪S_{p}}^{S_{p}})_{G(K_{S_{0}∪Sp}^{Sp}|k)}^{ab} ⊆ G(K_{S_{1}∪S_{p}}^{S_{p}}|K)_{G(K|k)}^{ab}.

Thus the abelian extension L|K with G(L|K) = G(K_{S_{1}∪S_{p}}^{S_{p}}|K)_{G(K|k)}^{ab} has the desired properties and we proved the claim.

As L|K is central over k, G(k|k^{+}) =< σ > acts on G(L|K). Thus G(L|K) = G(L|K)^{+} ⊕ G(L|K)^{−}, where G(L|K)^{±} = G(L|K)^{σ±1}. Let L^{±} be defined by G(L|L^{±}) = G(L|K)^{±}, i.e. G(L^{±}|K) = G(L|K)/G(L|K)^{∓}.

For 𝔮 ∈ T_{0}\S_{p} we have the exact sequence

1 ⟶ G_{𝔮}(L^{−}|k) ⟶ G_{𝔮}(L^{−}|k^{+}) ⟶ G_{𝔮}(k|k^{+}) ⟶ 1,

where G_{𝔮}(L^{−}|k) = G_{𝔮}(L^{−}|K). Suppose that G_{𝔮}(L^{−}|K) ≠ 1. Since G(k|k^{+}) = G_{𝔮}(k|k^{+}) acts non-trivially on G(L^{−}|K), the group G_{𝔮}(L^{−}|k^{+}) is non-abelian. On the other hand the extension L^{−}|k^{+} is unramified at 𝔮. This contradiction shows that all primes of T_{0}\S_{p} are completely decomposed in L^{−}|K, and so in L^{−}|k. Since L|K is completely decomposed at S_{p}, we obtain L^{−} ⊆ k^{T_{0}}.

Let 𝔭 ∈ V and so 𝔭 ∩ k^{+} splits in k|k^{+}. Let 𝔭̄ = 𝔭^{σ} be the conjugated prime and let 𝔓 be a prolongation of 𝔭 to K and 𝔓̄ = 𝔓^{σ}. By the claim it follows that

(T (k_{𝔓}(p))|K_{𝔓})_{G_{𝔓}(k(p))|k)} ⊕ T (k𝔓̄(p))|K𝔓̄)_{G𝔓̄(k(p))|k)})^{−}

injects into G(L|K)^{−} →^{∼} G(L^{−}|K) = G(L|K)/G(L|K)^{+}, and so L^{−}|K is ramified by 𝔓.

Varying the set S_{0} and the extension K|k, it follows that the extension k^{T_{0}}|k realizes the ramified part of k_{𝔭}(p)|k_{𝔭} for all 𝔭 ∈ S_{1} = P\T_{0}. Since k^{T_{0}}|k is a Galois extension, also the unramified part must be realized, i.e.

(k^{T_{0}})_{𝔓} = k_{𝔭}(p) for all 𝔓|𝔭, 𝔭 ∉ T_{0}

(if k_{𝔓}(p)|(k^{T_{0}})_{𝔓} would have a non-trivial unramified part, then, as the subgroup generated by the Frobenius automorphism is not normal, this extension would also have a ramified part).

The last assertion of the theorem follows from theorem (2.3). □

Let

∏ ∐_{t∈T}(A_{t}, B_{t}) = {(a_{t})_{t∈T} ∈ ∏_{t∈T} A_{t} | a_{t} ∈ B_{t} for almost all t ∈ T }

be the restricted product over a discrete set T of abelian locally compact groups A_{t} with respect to closed subgroups B_{t}. The topology is given by the subgroups V with

(i) V ∩ A_{t} is open in A_{t} for all t ∈ T, (ii) V ⊇ B_{t} for almost all t ∈ T. Then we call

∏ ∐_{t∈T}^{c}(A_{t}, B_{t}) := lim_{⟵}_{V} (∏ ∐_{t∈T}(A_{t}, B_{t}))/V,

the compactification of ∏ ∐_{t∈T} (A_{t}, B_{t}), where V runs through all open subgroups of finite index in ∏ ∐_{t∈T} (A_{t}, B_{t}). The the canonical map ∏ ∐_{t∈T} (A_{t}, B_{t}) → ∏ ∐_{t∈T}^{c} (A_{t}, B_{t}) has dense image.

We define the discretization of ∏ ∐_{t∈T} (A_{t}, B_{t}) by

∏ ∐_{t∈T}^{d}(A_{t}, B_{t}) := \underrightarrow{lim}_{W} W

where W runs through the finite subgroups of ∏ ∐_{t∈T} (A_{t}, B_{t}). If the subgroups B_{t} of A_{t}, t ∈ T, are open and compact, then ∏ ∐_{t∈T} (A_{t}, B_{t}) is locally compact. Using the equality

(∏ ∐_{t∈T}(A_{t}, B_{t}))^{∨} = ∏ ∐_{t∈T}(A_{t}^{∨}, (A_{t}/B_{t})^{∨}),

we obtain

∏ ∐_{t∈T}^{d}(A_{t}, B_{t}) = (((∏ ∐_{t∈T}(A_{t}, B_{t}))^{∨})^{c})^{∨} and ∏ ∐_{t∈T}^{c}(A_{t}, B_{t}) = (((∏ ∐_{t∈T}(A_{t}, B_{t}))^{∨})^{d})^{∨},

where ^{∨} denotes the Pontryagin-dual.

Proposition 2.5 Let T be a set of primes of a number field k with δ(T) = 0. Then

(i) G(k(p)/k^{T})/G(k(p)/k^{T})^{∗} = lim_{⟵}_{K} ∏ ∐_{𝔓∈T (K)}^{c}(G_{𝔓}(k)/G_{𝔓}(k)^{∗}, T̃_{𝔓}(k)),

(ii) H^{1}(G(k(p)/k^{T})) = \underrightarrow{lim}_{K} ∏ ∐_{𝔓∈T (K)}^{d}(H^{1}(G_{𝔓}(k)), H_{nr}^{1}(G_{𝔓}(k))),

where K runs through the finite subextensions of k^{T} |k and T̃_{𝔓}(k) is the group T_{𝔓}(k)G_{𝔓}(k)^{∗}/G_{𝔓}(k)^{∗}.

Proof: Since the set P\T has density equal to 1, we get from the Poitou-Tate exact sequence and the Hasse principle the commutative and exact diagram

[figure: 0; H^{1}(k̄|k, μ_{p}); ∏ ∐_{𝔭∉T}H^{1}(k̄_{𝔭}|k_{𝔭}, μ_{p}); 0; H^{1}(k̄|k, μ_{p}); ∏ ∐_{𝔭}H^{1}(k̄_{𝔭}|k_{𝔭}, μ_{p}); H^{1}(k̄|k, ℤ/pℤ)^{∨}; 0; ∏ ∐_{𝔭∈T}H^{1}(k̄_{𝔭}|k_{𝔭}, μ_{p}); ∼; ∏ ∐_{𝔭∈T}H^{1}(k̄_{𝔭}|k_{𝔭}, ℤ/pℤ)^{∨}]

where we use the local duality theorem

H^{1}(k̄_{𝔭}|k_{𝔭}, μ_{p}) ⟶^{∼} H^{1}(G_{𝔭}(k), ℤ/pℤ)^{∨} = G_{𝔭}(k)/G_{𝔭}(k)^{∗}

and for 𝔭 not above p or ∞ the isomorphism

H_{nr}^{1}(k̄_{𝔭}|k_{𝔭}, μ_{p}) ⟶^{∼} (H^{1}(T_{𝔭}(k), ℤ/pℤ)^{G_{𝔭}(k)})^{∨} ≅ T_{𝔭}(k)G_{𝔭}(k)^{∗}/G_{𝔭}(k)^{∗},

see [3] (8.6.10), (9.1.9), (9.1.10), (7.2.15). Thus we get a continuous injection

∏ ∐_{𝔭∈T (k)}(G_{𝔭}(k)/G_{𝔭}(k)^{∗}, T̃_{𝔭}(k)) ↪ G(k(p)/k)/G(k(p)/k)^{∗}

and so an injection ∏ ∐_{𝔭∈T (k)}^{c}(G_{𝔭}(k)//G_{𝔭}(k)^{∗}, T̃_{𝔭}(k)) ↪ G(k(p)/k)/G(k(p)/k)^{∗}. Passing to the limit, we obtain the injection

lim_{⟵}_{K} ∏ ∐_{𝔓∈T (K)}^{c}(G_{𝔓}(k)/G_{𝔓}(k)^{∗}, T̃_{𝔓}(k)) ↪ G(k(p)/k^{T})/G(k(p)/k^{T})^{∗}

which, by definition of the field k^{T}, is also surjective. Therefore we obtain (i) and (ii) is just the dual assertion. □

Proposition 2.6 Let p be a prime number and let T ≠ P be a stably saturated set of primes of a number field k. Assume that k is totally imaginary if p = 2. Then

cd_{p} G(k^{T} |k) ≤ 2.

Proof: We consider the Hochschild-Serre spectral sequence

E_{2}^{ij} = H^{i}(G/N, H^{j}(N)) ⇒ E^{i+j} = H^{i+j}(G)

for an extension 1 → N → G → G/N → 1 of pro-p groups. If cd_{p}(G) ≤ 2, E_{2}^{11} = 0 and the edge homomorphism E^{2} → E_{2}^{02} is surjective, then it follows that E_{2}^{30} = 0, i.e. cd_{p}(G/N) ≤ 2.

Now let G = G(k(p)|k) and N = G(k(p)|k^{T}). Using (2.2) and the Poitou-Tate theorem, it follows that the canonical map

H^{2}(k(p)|k) ↠ ⨁_{𝔭∈T (k)}H^{2}(G_{𝔭}(k)) ⟶^{∼} H^{0}(k^{T} |k, H^{2}(k(p)|k^{T}))

is surjective. Furthermore, using (2.5)(ii), we see that

H^{1}(k(p)|k^{T}) = \underrightarrow{lim}_{K} H^{1}(∏ ∐_{𝔭∈T (K)}^{c}(G_{𝔭}(k)/G_{𝔭}(k)^{∗}, T̃_{𝔭}(k)))
= \underrightarrow{lim}_{K} Coind^{G(K|k)} H^{1}(∏ ∐_{𝔭∈T (k)}^{c}(G_{𝔭}(k)/G_{𝔭}(k)^{∗}, T̃_{𝔭}(k)))
= Coind^{G(k^{T} |k)} H^{1}(∏ ∐_{𝔭∈T (k)}^{c}(G_{𝔭}(k)/G_{𝔭}(k)^{∗}, T̃_{𝔭}(k)))

is a G(k^{T} |k)-coinduced module, hence H^{1}(k^{T} |k, H^{1}(k(p)|k^{T})) = 0. Therefore we obtain the desired result. □

If T is a set of primes of k, then let T (k^{T}) = lim_{⟵}_{K} T (K) ∪ {∗_{K}}, where K

runs through the finite subextensions of k^{T} /k. The following theorem asserts that the Galois group G(k(p)/k^{T}), where T ≠ P is stably saturated, is a corestricted free pro-p-product of the family (G_{𝔓}(k))_{𝔓∈T (k^{T})} of decomposition groups with respect to the continuous family (T_{𝔓}(k))_{𝔓∈T (k^{T})} of inertia groups, see [2] for the definition.

Theorem 2.7 Let T ≠ P be a stably saturated set of primes of a number field k. Then the canonical map

φ : ∗_{𝔓∈T (k^{T})}(G_{𝔓}(k), T_{𝔓}(k)) ⟶^{∼} G(k(p)/k^{T})

is an isomorphism.

Proof: We can apply [2] prop. (4.3) and have to show that the induced maps

φ_{∗} : H^{1}(G(k(p)/k^{T}), ℤ/pℤ) →^{∼} \underrightarrow{lim}_{K} ∏ ∐_{T (K)}^{d}(H^{1}(G_{𝔓}(k), ℤ/pℤ), H_{nr}^{1}(G_{𝔓}(k), ℤ/pℤ))

φ_{∗} : H^{2}(G(k(p)/k^{T}), ℤ/pℤ) ↪ \underrightarrow{lim}_{K} ∏ ∐_{T (K)}^{d}(H^{2}(G_{𝔓}(k), ℤ/pℤ), H_{nr}^{2}(G_{𝔓}(k), ℤ/pℤ))

are bijective resp. injective. Since H_{nr}^{2}(G_{𝔓}(k)) = 0, it follows that

H^{2}(G) = \underrightarrow{lim}_{K} ∏ ∐_{T (K)}^{d}(H^{2}(G_{𝔓}(k), H_{nr}^{2}(G_{𝔓}(k))) = \begin{matrix}⨁ & ′ \\ 𝔭∈T (k^{T}) \end{matrix}H^{2}(G_{𝔭}(k)).

Now the result follows from (2.5)(ii) and (2.2). □

In the situation of theorem (2.4) we obtain

Corollary 2.8 Let p be an odd prime number and let k be a CM-field containing the group μ_{p} of all p-th roots of unity, with maximal totally real subfield k^{+}, i.e. k = k^{+}(μ_{p}) is totally imaginary and [k : k^{+}] = 2. Let

T = {𝔭 | 𝔭 ∩ k^{+} is inert in k|k^{+}} ∪ S_{p} .

Then

∗_{𝔓∈T (k^{T})}(G_{𝔓}(k), T_{𝔓}(k)) ⟶^{∼} G(k(p)/k^{T}).

References

[1] Fried, M.D., Haran, D., Völklein, H. Absolute Galois group of the totally real numbers. C.R. Acad. Sci., Paris 317 (1993) 995-999

[2] Gärtner, Jochen., Wingberg, K. Corestricted free products of profinite groups. Preprint 2013

[3] Neukirch, J., Schmidt, A., Wingberg, K. Cohomology of Number Fields, 2nd edition. Springer 2008

[4] Pop, F. Embedding problems over large fields. Ann. of Math. 144 (1996) 1-34

Mathematisches Institut der Universität Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany e-mail: wingberg (at) mathi.uni-heidelberg.de