DETECTING PRO-p-GROUPS THAT ARE NOT ABSOLUTE GALOIS GROUPS

DAVE BENSON, NICOLE LEMIRE, JÁN MINÁČ, AND JOHN SWALLOW

Let p be a prime. It is a fundamental problem to classify the absolute Galois groups G_{F} of fields F containing a primitive pth root of unity ξ_{p}. In this paper we present several constraints on such G_{F}, using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these G_{F}. In the case p > 2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. In section 2 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of G_{F}. Finally, in section 3 we construct a new family of pro-p-groups which are not absolute Galois groups over any field F.

As a consequence of our results, we prove the following limitations on relator shapes of pro-p absolute Galois groups. For elements σ and τ of a group Γ, let ^{0}[σ, τ] = τ, ^{1}[σ, τ] = στσ^{−1}τ^{−1}, and ^{n}[σ, τ] = [σ,^{n−1}[σ, τ]] for n ≥ 2. Similarly, for subsets Γ_{1} and Γ_{2} of Γ, let ^{n}[Γ_{1}, Γ_{2}] denote the closed subgroup generated by all elements of the form ^{n}[γ_{1}, γ_{2}] for γ_{i} ∈ Γ_{i}.

Theorem A.1. Let p be an odd prime, Γ a pro-p-group with maximal closed subgroup Δ, and σ ∈ Γ \ Δ.

(1) Suppose that for some τ ∈ Δ and some 2 ≤ e ≤ p − 2

e[σ, τ] ∉ ^{p−1}[σ, Δ]Φ(Δ) and ^{e+1}[σ, τ] ∈ Φ(Δ).

Then Γ is not an absolute Galois group.

Date: October 20, 2006.

Nicole Lemire was supported in part by Natural Sciences and Engineering Research Council of Canada grant R3276A01. Ján Mináč was supported in part by Natural Sciences and Engineering Research Council of Canada grant R0370A01.

John Swallow was supported in part by National Science Foundation grant DMS-0600122.

2 DAVE BENSON, NICOLE LEMIRE, JÁN MINÁČ, AND JOHN SWALLOW

Moreover, if Γ contains a normal subgroup Λ ⊂ Δ such that Γ/Λ ≃ ℤ/p^{2}ℤ, we may take 1 ≤ e ≤ p − 2.

(2) Suppose that for some τ_{1}, τ_{2} ∈ Δ,

[σ, τ_{i}] ∉ ^{p−1}[σ, Δ]Φ(Δ), ^{2}[σ, τ_{i}] ∈ Φ(Δ), i = 1, 2

⟨[σ, τ_{1}]⟩Φ(Δ) ≠ ⟨[σ, τ_{2}]⟩Φ(Δ)

Then Γ is not an absolute Galois group.

(3) Suppose that

σ^{p} ∈ ^{2}[σ, Δ]Φ(Δ).

Then Γ is not an absolute Galois group.

Here Φ(Δ) = Δ^{p}[Δ, Δ] denotes the Frattini subgroup of Δ.

Furthermore, pro-p-groups with single relations similar to those of Demuškin groups for odd primes cannot be absolute Galois groups.

Corollary. Let p be an odd prime and Γ a pro-p-group minimally generated by {σ_{1}, σ_{2}} ∪ {σ_{i}}_{i∈I} subject to a single relation

σ_{1}^{q} ·^{f}[σ_{1}, σ_{2}] · ∏_{(i,j)∈J}[σ_{i}, σ_{j}] · ∏_{k∈K}[σ_{1}^{p}, σ_{k}]

for some 2 ≤ f ≤ p − 1, q ∈ ℕ ∪ {0} with q = 0 mod (p^{2}), a finite ordered set of pairs J ⊂ I × I, and a finite ordered subset K of I. Then Γ is not an absolute Galois group.

The results in [LMS] may be used to establish further new results on possible V -groups of fields and metabelian quotients of absolute Galois pro-p-groups. (For the definition of the V -group V_{F} of a field F, see section 2.) Moreover, some of the results here hold in a greater generality than their formulations here. For instance, the examples here of pro-p non-absolute Galois groups are also examples of groups which are not maximal pro-p-quotients of absolute Galois groups, by [LMS, §6]. Furthermore, pro-p-groups which are not absolute Galois groups are not p-Sylow subgroups of absolute Galois groups. We plan a systematic study of these concerns in [BMS].

We observe that because this paper is concerned only with degree 1 and degree 2 cohomology, the results cited from [LMS] rely only on the Merkurjev-Suslin Theorem [MeSu, Theorem 11.5] and not the full Bloch-Kato Conjecture. Furthermore, we note that while this paper is self-contained, an extended version is available [BLMS].

DETECTING PRO-p-GROUPS 3

1. T -groups

A T -group is a nontrivial pro-p-group T with a maximal closed subgroup N that is abelian of exponent dividing p. Then N is a normal subgroup, and the factor group T/N acts naturally on N: choose a lift t ∈ T and act via n ↦ tnt^{−1}. Given any profinite group Γ with a closed normal subgroup Δ of index p, the factor group Γ/Δ^{p}[Δ, Δ] is a T -group. Now suppose that E/F is a cyclic field extension of degree p. We define the T -group of E/F to be T_{E/F} := G_{F} /G_{E}^{p}[G_{E}, G_{E}]. In this section we classify those T -groups realizable as T_{E/F} for fields F either with char F = p or ξ_{p} ∈ F.

We develop a complete set of invariants t_{i}, i = 1, 2, …, p, and u of T -groups as follows. For a pro-p-group Γ, denote by Z(Γ) its center, Z(Γ)[p] the elements of Z(Γ) of order dividing p, Γ^{(n)} the nth group in the p-central series of Γ, and Γ_{(n)} the nth group in the central series of Γ. For a T -group T we define

t_{1} = dim_{𝔽_{p}} H^{1} (\frac{Z(T)[p]}{Z(T) ∩ T_{(2)}} , 𝔽_{p})
t_{i} = dim_{𝔽_{p}} H^{1} (\frac{Z(T) ∩ T_{(i)}}{Z(T) ∩ T_{(i+1)}} , 𝔽_{p}) , 2 ≤ i ≤ p
u = max{i : 1 ≤ i ≤ p, T^{p} ⊂ T_{(i)}}.

Proposition A.1. For arbitrary cardinalities t_{i}, i = 1, 2, …, p, and u with 1 ≤ u ≤ p, the following are equivalent:

(1) The t_{i} and u are invariants of some T -group

(2) (a) if u < p then t_{u} ≥ 1, and (b) if u = p and t_{i} = 0 for all 2 ≤ i ≤ p, then t_{1} ≥ 1.

Moreover, T -groups are uniquely determined up to isomorphism by these invariants.

Theorem A.2. For p an odd prime, the following are equivalent:

(1) T is a T -group with invariants t_{i} and u satisfying (a) u ∈ {1, 2},

(b) t_{2} = u − 1, and

(c) t_{i} = 0 for 3 ≤ i < p.

(2) T ≃ T_{E/F} for some cyclic extension E of degree p of a field F such that either char F = p or ξ_{p} ∈ F.

Now suppose p = 2. Then each T -group is isomorphic to T_{E/F} for some cyclic extension E/F of degree 2.

Let G be a cyclic group of order p and M_{i}, i = 1, 2, …, p, denote the unique cyclic 𝔽_{p}G-module of dimension i. Since the 𝔽_{p}G-modules we consider will be multiplicative groups, we usually write the action of G exponentially. For a set I, let M_{i}^{I} denote the product of |I| copies of M_{i} endowed with the product topology. We use the word duality exclusively to refer to Pontrjagin duality, between compact and discrete abelian groups and more generally between compact and discrete 𝔽_{p}Gmodules. We denote the dual of Γ by Γ^{∨}.

Lemma A.1. Suppose T is a T -group with invariants t_{i}, i = 1, 2, …, p and u, and N is a maximal closed subgroup of T that is abelian of exponent dividing p. Let σ ∈ T \ N and set σ̄ to be the image of σ in G := T/N. Then N is a topological 𝔽_{p}G-module and we have

(1) For any 𝔽_{p}G-submodule M of N and i ≥ 0, ^{i}[T, M] = M^{(σ̄−1)^{i}}.

(2) For i ≥ 2, T_{(i)} = N^{(σ̄−1)^{i−1}}.

(3) There exist sets I_{i}, i = 1, 2, …, p, such that N decomposes into indecomposable 𝔽_{p}G-modules as N = M_{1}^{I_{1}} × M_{2}^{I_{2}} × ⋯ × M_{p}^{I_{p}}, endowed with the product topology. Moreover, for i ≥ 2, t_{i} = |I_{i}|, and

t_{1} = \begin{cases}1 + |I_{1}|, & T is abelian of exponent p \\ |I_{1}|, & otherwise. \end{cases}

(4) T^{p} = ⟨σ^{p}⟩ · T_{(p)}.

(5) If u < p then u is the minimal v with 1 ≤ v ≤ p − 1 such that there is a commutative diagram of pro-p-groups

\xymatrix{1 \ar[r] & N \ar[r] \ar@{->>}[d] & T \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{v} \ar[r] & H \ar[r] & G \ar[r] & 1}  (1)

with a lift of σ̄ in H of order p^{2}. If u = p then no such diagram exists for 1 ≤ v ≤ p − 1.

Proof. (1). Suppose that τ ∈ T is arbitrary, and write τ = nσ^{i} for n ∈ N and i ∈ ℕ∪{0}. The action of T factors through G. Hence [τ, M] = M^{(σ̄^{i}−1)} and so [T, M] = M^{(σ̄−1)}. The result follows by induction. (Moreover, we observe that if p does not divide i, then [τ, M] = [M, τ] = [T, M] = [M, T].)

(2). Observe that [T, T] = [T, N]. Then use (1).

(3). Because 𝔽_{p}G is an Artinian principal ideal ring, every 𝔽_{p}Gmodule U decomposes into a direct sum of cyclic 𝔽_{p}G-modules. Every cyclic 𝔽_{p}G-module is indecomposable and self-dual. Applying these results to U = N^{∨} and using duality (see [RZ, Lemma 2.9.4 and Theorem 2.9.6]), we obtain the decomposition.

Using (2) together with M_{j}^{G} = M_{j}^{(σ̄−1)^{j−1}} for all 1 ≤ j ≤ p, we calculate

Z(T) ∩ N = (M_{1})^{I_{i}} × (M_{2}^{(σ̄−1)})^{I_{2}} × ⋯ × (M_{p}^{(σ̄−1)^{p−1}})^{I_{p}} ,
Z(T) ∩ T_{(i)} = (M_{i}^{(σ̄−1)^{i−1}})^{I_{i}} × ⋯ × (M_{p}^{(σ̄−1)^{p−1}})^{I_{p}} ,  2 ≤ i ≤ p,

and Z(T) ∩ T_{(i)} = {1} for i > p. We deduce that t_{i} = |I_{i}|, 2 ≤ i ≤ p.

For the case i = 1, suppose first that T is abelian of exponent p. Then Z(T) = Z(T)[p] = T and T_{(2)} = {1}. By (2), N^{(σ̄−1)} = {1}, whence |I_{i}| = 0 for i ≥ 2. Therefore t_{1} = 1 + |I_{1}|. Next suppose that T is nonabelian. Then Z(T) ⊂ N. We obtain Z(T)[p] = Z(T) ∩ N and so t_{1} = dim_{𝔽_{p}} H^{1} ((Z(T) ∩ N)/(Z(T) ∩ T_{(2)}), 𝔽_{p}) = |I_{1}|. Finally, assume that T is abelian and not of exponent p. Then N = Z(T)[p] and t_{1} = dim_{𝔽_{p}} H^{1}(N, 𝔽_{p}) = |I_{1}|.

(4). For δ ∈ N we have (δσ)^{2} = [σ, δ]δ^{2}σ^{2} and, by induction,

(δσ)^{i} = [\underbrace{σ, [σ, …, [σ}_{i−1 times}, δ] ⋯]]^{(\begin{matrix}i \\ i \end{matrix})} ⋯ [σ, [σ, δ]]^{(\begin{matrix}i \\ 3 \end{matrix})}[σ, δ]^{(\begin{matrix}i \\ 2 \end{matrix})}δ^{i}σ^{i}.

Then (Nσ)^{p} = (σ^{p}) · ^{p−1}[σ, N], which by (1) and (2) may be written σ^{p}·T_{(p)}. Replacing σ with σ^{v} for (v, p) = 1, we conclude T^{p} = ⟨σ^{p}⟩·T_{(p)}.

(5). Suppose that for some v < u there is a commutative diagram (1) with a lift of σ̄ in H of order p^{2}. Then T ↠ H factors through T/N^{(σ̄−1)^{v}}. But by (2), T_{(v+1)} = N^{(σ̄−1)^{v}} and by definition of u, we have T^{p} ⊂ T_{(u)} ⊂ T_{(v+1)}. Hence every lift of σ̄ into T/N^{(σ̄−1)^{v}} is of order p, and the same holds for H. We conclude that no commutative diagram as above with σ̄ lifting to an element of order p^{2} exists for v < u.

Now suppose that u < p and consider σ^{p}. By (4), T^{p} = ⟨σ^{p}⟩ · T_{(p)}, and T_{(p)} ⊂ T_{(u+1)}. We have T^{p} ⊂ T_{(u)} and T^{p} ⊄ T_{(u+1)} and so σ^{p} ∈ T_{(u)} \ T_{(u+1)}. Now σ^{p} ∈ N. By (2), since T_{(u)} = N^{(σ−1)^{u−1}} for u ≥ 2, we deduce σ^{p} ∈ N^{(σ−1)^{u−1}} \ N^{(σ−1)^{u}}. From [σ, σ^{p}] = 1 we obtain σ^{p} ∈ N^{G}.

Therefore σ^{p} ∈ (N^{G} ∩ N^{(σ−1)^{u−1}}) \ (N^{G} ∩ N^{(σ−1)^{u}}). We claim that there exists an 𝔽_{p}G-submodule M_{u} of N such that M_{u}^{G} = ⟨σ^{p}⟩ and N = M_{u} ×Ñ for some 𝔽_{p}G-submodule Ñ of N. Assume a factorization of N into cyclic 𝔽_{p}G-submodules as in (3). Consider w ∈ N such that w^{(σ−1)^{u−1}} = σ^{p} and all components of w lie in factors of dimension at least u. For at least one factor M_{u}, proj_{M_{u}} w generates M_{u} as an 𝔽_{p}G-module. Write N = M_{u}×Ñ. We obtain the commutative diagram

\xymatrix{1 \ar[r] & N \ar[r] \ar@{->>}[d] & T \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{u} \ar[r] & T/Ñ \ar[r] & G \ar[r] & 1}

in which a lift of σ̄ is of order p^{2}. □

The following lemma follows easily from the definitions.

Lemma A.2. Let T be a T -group with invariants t_{i}, i = 1, 2, …, p, and u.

(1) T is abelian if and only if t_{i} = 0 for all 2 ≤ i ≤ p.

(2) T is of exponent p if and only if u = p and t_{u} = 0.

(3) If u < p then t_{u} ≥ 1.

(4) If t_{i} = 0 for all 2 ≤ i ≤ p, then t_{1} ≥ 1.

Proof of Proposition A.1. By Lemma A.2(3,4), (1) implies (2). Now suppose we are given cardinalities t_{i}, i = 1, 2, …, p, and u satisfying conditions (2).

The case u < p. Let G be a group of order p and I_{i}, i = 1, 2, ⋯, p sets with cardinalities |I_{i}| satisfying |I_{i}| = t_{i} for i ≠ u and 1+|I_{u}| = t_{u}. Set N = X ×M_{1}^{I_{1}} ×M_{2}^{I_{2}} ×⋯×M_{p}^{I_{p}}, where X ≃ M_{u} and N is endowed with the product topology. Then N is a pseudocompact 𝔽_{p}G-module. (See [Br, page 443].) Define an action of ℤ_{p} on N by letting a generator σ of ℤ_{p} act via a generator of G, and form N ⋊ ℤ_{p} in the category of pro-p-groups. Now choose an 𝔽_{p}G-module generator x of X and define x_{i} = x^{(σ−1)^{i}} for 0 ≤ i ≤ u. Since (σ − 1) is nilpotent of degree u on X we obtain x_{u−1} ≠ 1 and x_{u} = 1. We set R to be the closed subgroup ⟨σ^{p}x_{u−1}⟩ ⊂ N ⋊ ℤ_{p}. Finally form Γ = (N ⋊ ℤ_{p})/R and set Δ to be the image of N ⋊ {1} in Γ. Since Δ ≃ N as pro-p-G operator groups, we identify them. By construction Δ is a maximal closed subgroup of Γ which is abelian of exponent p. Hence Γ is a T -group. Since the image of σ in Γ has order p^{2}, Γ is not of exponent p. From the decomposition of N, we obtain by Lemma A.1(3) that the invariants t_{i} are as desired. It remains only to show that u is as given. By Lemma A.1(4) we have Γ^{p} = ⟨x_{u−1}⟩ · Γ_{(p)}. From Lemma A.1(1,2) we calculate that x_{u−1} ∈ Γ_{(u)} and x_{u−1} ∉ Γ_{(u+1)}. Hence u is as desired.

The case u = p follows analogously. Let G be a group of order p and I_{i} sets with cardinalities |I_{i}| satisfying |I_{i}| = t_{i}, 2 ≤ i ≤ p; |I_{1}| = t_{1} if some t_{j} ≠ 0, 2 ≤ j ≤ p; and 1 + |I_{1}| = t_{1} if all t_{j} = 0, 2 ≤ j ≤ p. Set N = M_{1}^{I_{1}} × M_{2}^{I_{2}} × ⋯ × M_{p}^{I_{p}}, Γ = N ⋊ G, Δ = N ⋊ {1}, let σ be a generator of G, and proceed as before.

Now we show that T -groups are uniquely determined up to isomorphism by the invariants t_{i} and u. Let T be an arbitrary T -group with invariants t_{i}, i = 1, 2, …, p, and u, N a maximal closed subgroup of T that is abelian of exponent dividing p, and G = T/N. From Lemma A.1(3) the structure of N as an 𝔽_{p}G-module is determined up to isomorphism, and T is an extension of N by G. Let σ ∈ T \ N. We have σ^{p} ∈ N^{G}. It remains only to determine the isomorphism class of N as an 𝔽_{p}G-module with a distinguished factor X such that σ^{p} ∈ X^{G}.

Suppose first that u = p, t_{1} ≥ 1, and t_{i} = 0 for all 2 ≤ i ≤ p. From Lemma A.2(1), T is abelian and so T_{(2)} = {1}. Since u = p, T has exponent p. Then T ≃ M_{1}^{I_{1}} ×G and t_{1} = |I_{1}|+1. Thus T is determined by the invariants. Now suppose that t_{1} ≥ 1, t_{i} = 0 for all 2 ≤ i ≤ p, and u = 1. Again T is abelian. Since u ≠ p, T has exponent p^{2}. Then N = X × Ñ, where σ^{p} generates an 𝔽_{p}G-module X isomorphic to M_{1} and Ñ ≃ M_{1}^{I′} with |I′| + 1 = t_{1}.

Finally suppose t_{i} ≠ 0 for some i with 2 ≤ i ≤ p. Then T is nonabelian, and by Lemma A.1(4), T^{p} = ⟨σ^{p}⟩·T_{(p)}. From Lemma A.1(2) we obtain, for u < p, σ^{p} ∈ (N^{G} ∩N^{(σ−1)^{u−1}})\(N^{G} ∩N^{(σ−1)^{u}}), while for u = p, we have σ^{p} ∈ N^{(σ−1)^{p−1}}. If u < p then by Proposition A.1 we have t_{u} ≥ 1, and using the same argument as in the proof of Lemma A.1(5), N contains a distinguished direct factor X ≃ M_{u} such that X^{G} = ⟨σ^{p}⟩. We deduce that N = X × Ñ where Ñ ≃ M_{u}^{I′} × ∏_{i≠u} M_{i}^{I_{i}} for sets I_{i}, i ≠ u, and I′ such that |I_{i}| = t_{i}, i ≠ u, and 1 + |I′| = t_{u}. If u = p, we claim that without loss of generality we may assume that σ^{p} = 1. Since σ^{p} ∈ N^{(σ−1)^{p−1}}, let ν ∈ N such that σ^{p} = ν^{(σ−1)^{p−1}} and set τ = σν^{−1}. Then τ ∈ T \ N and τ^{p} = σ^{p}(ν^{−1})^{1+σ+···+σ^{p−1}} = 1. Hence T = N ⋊ G. □

Lemma A.3. Suppose that Γ is a profinite group such that its maximal pro-p-quotient Γ(p) is a free pro-p-group of (positive and possibly infinite) rank n, and let Δ be a normal subgroup of Γ of index p. Then the invariants of the T -group Γ/Δ^{p}[Δ, Δ] are t_{1} = 1, t_{i} = 0 for 2 ≤ i < p, t_{p} = n − 1 if n < ∞ and t_{p} = n for n an infinite cardinal, and u = 1.

Proof. Since Γ/Δ^{p}[Δ, Δ] = Γ(p)/Φ(Δ(p)), we may assume without loss that Γ is a free pro-p-group. The result then follows from the analogue of the Kurosh subgroup theorem in the context of pro-p-groups. □

Lemma A.4. Let S be a free pro-p-group. Then there exists a field F of characteristic 0 such that G_{F} ≃ S.

Proof. First let F_{0} be any algebraically closed field of characteristic 0 with cardinality greater than or equal to d = dim_{𝔽_{p}} H^{1}(S, 𝔽_{p}). Set F_{1} := F_{0}(t). By [Do], G_{F_{1}} is a free profinite group, and let P denote a p-Sylow subgroup of G_{F_{1}}. By [RZ, Corollary 7.7.6], P is a free pro-p-group. Let F_{2} be the fixed field of P. The classes in F_{1}^{×}/F_{1}^{×p} of the set A of linear polynomials t − c, c ∈ F_{0}, are linearly independent over 𝔽_{p}. Choose a subset of A of cardinality d, and let V be the vector subspace of F_{1}^{×}/F_{1}^{×p} generated by the classes of the elements of A. Since ([F_{2} : F_{1}], p) = 1, V injects into F_{2}^{×}/F_{2}^{×p}. Let W denote this image. Now let F be a maximal algebraic field extension of F_{2} such that W injects into F^{×}/F^{×p}. By maximality, the image i(W) of W in F^{×}/F^{×p} is in fact F^{×}/F^{×p}. The rank of G_{F} is then dim_{𝔽_{p}} H^{1}(G_{F} , 𝔽_{p}) = dim_{𝔽_{p}} V = d. □

Proof of Theorem A.2. The case p = 2. Let u ∈ {1, 2}, t_{1}, and t_{2} be invariants of a T -group T. By Proposition A.1, t_{1} ≥ 1 if u = 1, and if u = 2 then either t_{1} ≥ 1 or t_{2} ≥ 1.

Case 1 : T is not of exponent 2. By Lemma A.2(2), either u = 1 or t_{2} ≥ 1. Let G be a group of order 2 and N = M_{1}^{I_{1}} ×M_{2}^{I_{2}} for sets I_{1} and I_{2} satisfying |I_{1}| = t_{1} and |I_{2}| = t_{2}, and let M = N^{∨}. From [MSw, Corollary 2], there exists E/F with char F ≠ 2 and Gal(E/F) ≃ G such that H^{1}(G_{E}, 𝔽_{2}) ≃ E^{×}/E^{×2} ≃ M as 𝔽_{2}G-modules if and only if there exist Υ ∈ {0, 1} and cardinalities d and e such that t_{1}+1 = 2Υ+d; t_{2} + Υ = e; if Υ = 0 then d ≥ 1; and if Υ = 1 then e ≥ 1. Moreover, −1 ∈ N_{E/F} (E^{×}) if and only if Υ = 1. Finally, by [MSw, proof of Theorem 1], we may choose E/F such that G_{F} is a pro-2-group.

If u = 1 then set Υ = 1 and e = t_{2} + Υ. Since t_{1} ≥ 1 we may choose d ≥ 0 such that 2Υ+d = t_{1}+1. Then e ≥ 1 and the conditions for E/F with G_{E}/G_{E}^{(2)} ≃ M^{∨} ≃ N are satisfied. Since Υ = 1, −1 ∈ N_{E/F} (E^{×}), and by Albert’s criterion [A], E/F embeds in a cyclic extension E′/F of degree 4. Let Gal(E/F) = ⟨σ̄⟩. We have the commutative diagram

\xymatrix{1 \ar[r] & G_{E}/G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G_{F} /G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{1} \ar[r] & H \ar[r] & G \ar[r] & 1}

in which σ̄ lifts to an element of order 4. By Lemma A.1(5), u = 1 for T_{E/F}. By Lemma A.2(2), T_{E/F} is not of exponent 2. Using Lemma A.1(3), because G_{E}/G_{E}^{(2)} ≃ N ≃ M_{1}^{I_{1}} × M_{2}^{I_{2}}, the invariants t_{1} and t_{2} of T_{E/F} match those of T. By Proposition A.1, T ≃ T_{E/F}.

If u = 2 then t_{2} ≥ 1. We take Υ = 0, d = t_{1} + 1 ≥ 1, and e = t_{2} and obtain an extension E/F as before. Since Υ = 0, −1 ∉ N_{E/F} (E^{×}) and so by [A], E/F does not embed in a cyclic extension E′/F of degree 4. Let Gal(E/F) = ⟨σ̄⟩. There is no commutative diagram

\xymatrix{1 \ar[r] & G_{E}/G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G_{F} /G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{1} \ar[r] & H \ar[r] & G \ar[r] & 1}

in which σ̄ lifts to an element of order 4. By Lemma A.1(5), u = 2 for T_{E/F}. Because t_{2} ≥ 1, G_{E}/G_{E}^{(2)} contains an 𝔽_{2}G-submodule isomorphic to M_{2} whence T_{E/F} is nonabelian. By Lemma A.1(3) and the isomorphism G_{E}/G_{E}^{(2)} ≃ N ≃ M_{1}^{I_{1}} × M_{2}^{I_{2}} we deduce that the invariants t_{1} and t_{2} of T_{E/F} match those of T. By Proposition A.1, T ≃ T_{E/F}.

Case 2 : T has exponent 2. By Lemma A.2(2), u = 2 and t_{2} = 0, and so t_{1} ≥ 1. Let N = M_{1}^{I_{1}} for I_{1} satisfying |I_{1}| + 1 = t_{1}. Set Υ = 0, d = t_{1} + 1, and e = t_{2} = 0. Then d ≥ 1 and there exists E/F with char F ≠ 2 such that H^{1}(G_{E}, 𝔽_{2}) ≃ M and −1 ∉ N_{E/F} (E^{×}). As before, u = 2 for T_{E/F}, and by Lemma A.1(3), t_{2} = 0 for T_{E/F}. By Lemma A.2(2), T_{E/F} has exponent 2, and by Lemma A.1(3), t_{1} is the correct invariant for T_{E/F}. By Proposition A.1, T ≃ T_{E/F}.

The case p > 2. First we characterize those T -groups occurring as T_{E/F} for fields F such that the maximal pro-p-quotient G_{F} (p) of the absolute Galois group G_{F} is free pro-p. Lemma A.3 tells us that for such F and an E/F of degree p, the invariants of T_{E/F} are t_{1} = 1, t_{i} = 0 for 2 ≤ i < p, and u = 1, and that the rank of G_{F} (p) is one more than the invariant t_{p}. Now suppose that T is a T -group with invariants t_{1} = 1, t_{i} = 0 for 2 ≤ i < p, and u = 1. By Lemma A.4 there exists F such that G_{F} is a free pro-p-group of rank t_{p} + 1. Letting Δ be any maximal closed subgroup of G_{F} and E = Fix(Δ), Lemma A.3 and Proposition A.1 give T ≃ T_{E/F}. Therefore the T -groups which occur as T_{E/F} for fields F with free maximal pro-p-quotient G_{F} (p) are precisely those for which t_{1} = 1, t_{i} = 0 for 2 ≤ i < p, and u = 1.

Now we characterize which of the remaining T -groups occur as T_{E/F} for cyclic field extensions E/F of degree p for F a field such that either char F = p or ξ_{p} ∈ F. By Witt’s Theorem, char F ≠ p. Hence we consider only fields F with char F ≠ p and ξ_{p} ∈ F. For a cyclic extension E/F of degree p, consider the 𝔽_{p} Gal(E/F)-module M_{E/F} := H^{1}(G_{E}, 𝔽_{p}) ≃ E^{×}/E^{×p}, and let G be an abstract group of order p. Since the particular isomorphism G ≃ Gal(E/F) does not alter the isomorphism class of M_{E/F} as an 𝔽_{p}G-module, we may consider all such modules 𝔽_{p}G-modules. (See [MSw].) By [MSw, Corollary 1], M ≃ M_{E/F} as 𝔽_{p}G-modules for suitable E/F with G ≃ Gal(E/F) if and only if M = M_{1}^{I_{1}} ⊕ M_{2}^{I_{2}} ⊕ M_{p}^{I_{p}}, where the cardinalities j_{1} = |I_{1}|, j_{2} = |I_{2}|, and j_{p} = |I_{p}| satisfy the following conditions: j_{1}+1 = 2Υ+d, j_{2} = 1 − Υ, and j_{p} + 1 = e for some cardinalities d, e and Υ ∈ {0, 1} where d ≥ 1 if Υ = 0 and e ≥ 1. Moreover, Υ = 1 if and only if ξ_{p} ∈ N_{E/F} (E^{×}). Finally, by [MSw, proof of Theorem 1], we may choose E/F such that G_{F} is a pro-p-group.

We observe that the constraints on j_{1}, j_{2}, and j_{p} are then j_{1} ≥ Υ and j_{2} = 1 − Υ. Now by duality, H^{1}(G_{E}, 𝔽_{p})^{∨} ≃ G_{E}/G_{E}^{(2)}, and since cyclic 𝔽_{p}G-modules are self-dual, we may derive conditions on the topological 𝔽_{p}G-module G_{E}/G_{E}^{(2)} occurring as maximal closed subgroups of T -groups G_{F} /G_{E}^{(2)}, as follows. Set G = G_{F} /G_{E}.

First we relate Υ and the invariant u. We claim that for any T -group G_{F} /G_{E}^{(2)}, we have u ≤ 2. Write E = F (\sqrt[p]{a}) for some a ∈ F^{×}. Let [e] ∈ E^{×}/E^{×p} denote the class of e ∈ E^{×}. Then X = ⟨[\sqrt[p]{a}], [ξ_{p}]⟩ is a cyclic 𝔽_{p}G-submodule of M and is isomorphic to M_{i} for some i ∈ \{1, 2\}. By equivariant Kummer theory (see [W]), L = E( \sqrt[p^{2}]{a}, ξ_{p^{2}}) is a Galois extension of F. Moreover, G_{F} /G_{L} is a homomorphic image of G_{F} /G_{E}^{(2)}, since L/E is an elementary abelian extension. Then L/F (ξ_{p^{2}}) is Galois with group ℤ/p^{2}ℤ, and for any σ ∈ G_{F} \ G_{E}, the restriction σ_{L} ∈ Gal(L/F) restricts to a generator. Hence 1 ≠ σ^{p} ∈ Gal(L/E), and therefore 1 ≠ σ^{p} ∈ G_{E}/G_{E}^{(2)}. We have a commutative diagram

\xymatrix{1 \ar[r] & G_{E}/G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G_{F} /G_{E}^{(2)} \ar[r] \ar@{->>}[d] & G \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & M_{i} \ar[r] & G_{F} /G_{L} \ar[r] & G \ar[r] & 1}  (2)

and so by Lemma A.1(5) we deduce that u ≤ i ≤ 2.

Now we claim that Υ = 1 if and only if u = 1. We have that Υ = 1 if and only if ξ_{p} ∈ N_{E/F} (E^{×}). By [A], Υ = 1 if and only if E/F embeds in a cyclic extension of F of degree p^{2}, if and only if there exists a closed normal subgroup Ñ ⊂ G_{E} such that G_{F} /Ñ ≃ ℤ/p^{2}ℤ. Any such closed normal subgroup must contain G_{E}^{(2)}. Hence we deduce that Υ = 1 if and only if there exists a commutative diagram (2) with i = 1 in which nontrivial elements of G lift to elements of order p^{2}. By Lemma A.1(5), Υ = 1 if and only if u = 1.

We have therefore shown that u ≤ 2 and u = 2 − Υ. Translating the remaining conditions on j_{1} and j_{2}, we see that j_{1} ≥ 2 − u and j_{2} = u − 1. Now by Lemma A.1(3), t_{2} = j_{2}, t_{i} = 0 for 3 ≤ i ≤ p − 1, and t_{p} = j_{p}. Moreover, t_{1} = j_{1} if T is not abelian of exponent p. By Lemma A.2(2), T is of exponent p if and only if u = p. But we have shown that u ≤ 2 < p, whence T is not of exponent p and we have t_{1} = j_{1}. By Proposition A.1, if u = 1 then t_{1} ≥ 1, and since t_{2} = u − 1 the conditions for applying [MSw, Corollary 1] are valid. Hence a T -group T_{E/F} with prescribed invariants subject to conditions (1) exists. □

Proof of Theorem A.1. Suppose Γ is a pro-p-group with maximal closed subgroup Δ, and let T = Γ/Δ^{(2)}, N = Δ/Δ^{(2)} and G = Γ/Δ. Write σ̄ and τ̄ for the images of σ and τ, respectively.

(1). Since Δ surjects onto N we have that σ̄ ∉ N, τ̄ ∈ N, ^{e}[σ̄, τ̄] ∉ p−1[σ̄, N], and ^{e+1}[σ̄, τ̄] = 1. By Lemma A.1(2,3) we have 1 ≠ ^{e}[σ̄, τ̄] ∈ T_{(e+1)} \ T_{(p−1)}. We deduce from ^{e+1}[σ̄, τ̄] = 1 that ^{e}[σ̄, τ̄] ∈ Z(T). We obtain that some invariant t_{i} ≠ 0 for 3 ≤ i < p. Now if Γ = G_{F} for some F, then setting E = Fix(Δ) we obtain T = T_{E/F}, contradicting Theorem A.2. Now assume additionally that there exists a closed normal subgroup Λ ⊂ Δ of Γ such that Γ/Λ ≃ ℤ/p^{2}ℤ and e = 1. Let σ̃ denote the image of σ in T. We have a commutative diagram (1) with v = 1 and an image of σ̃ in H of order p^{2}. By Lemma A.1(5), u = 1 for T is equal to 1. As before some t_{i} ≠ 0, 2 ≤ i < p. Again by Theorem A.2 we are done.

(2). We proceed as before, obtaining two elements [σ̄, τ̄_{i}], i = 1, 2, which generate distinct subgroups of Z(T)∩T_{(2)} with trivial intersection with T_{(p)}. We deduce that the sum of t_{i}, 2 ≤ i < p, is at least two, and we apply Theorem A.2.

(3). We obtain σ̄ ∉ N, τ̄ ∈ N, and σ̄^{p} ∈ ^{2}[σ̄, N]. By Lemma A.1(2,3) we have σ̄^{p} ⊂ T_{(3)}, and from Lemma A.1(4) we deduce that T^{p} ⊂ T_{(3)}. Hence u ≥ 3, and we apply Theorem A.2. □

Proof of Corollary to Theorem A.1. Let Δ be the closed subgroup of Γ generated as a normal subgroup by σ_{1}^{p}, σ_{2}, and σ_{i} for i ∈ I, and let Λ be the closed subgroup of Γ generated as a normal subgroup by σ_{1}^{p^{2}}, σ_{2}, and σ_{i} for i ∈ I. Set also T = Γ/Δ^{(2)}. Examining the quotient of Γ obtained by trivializing σ_{2} and each σ_{i}, i ∈ I, we see that Δ is maximal and Γ/Λ ≃ ℤ/p^{2}ℤ. Now let e = f − 1. By passing from Γ to T using bars for denoting images of elements of Γ in T, we see that e[σ̄_{1}, σ̄_{2}] ∉ ^{p−1}[σ̄_{1}, Δ̄] in T. On the other hand,^{e+1}[σ_{1}, σ_{2}] = ^{f}[σ_{1}, σ_{2}] = 1 in T. By Theorem A.1(1), Γ is not an absolute Galois group. □

It is a natural question to ask whether the maximal closed subgroup N in the definition of T -group is unique. It is not difficult to show that if T is a T -group with a maximal closed subgroup N which is abelian of exponent dividing p, then unless T is either itself abelian of exponent p of order greater than p, or isomorphic to the direct product of a Heisenberg group of order p^{3} and an abelian group of exponent dividing p, then N is a characteristic subgroup of T. Since these exceptional T -groups have invariant u = p, we have by Theorem A.2 that for p > 2, each T_{E/F} has a unique index p elementary abelian subgroup.

2. Analogues of Theorems of Artin-Schreier and Becker

Recall that by Artin-Schreier there are no elements of order p in the absolute Galois groups of a field F, unless p = 2 and F is formally real. Becker proved that the same holds for maximal pro-p-quotients of absolute Galois groups [Be]. In this section we show that in certain small quotients of absolute Galois groups, there are no non-central elements of order p unless p = 2 and the base field F is formally real.

Let p be a prime and F a field with ξ_{p} ∈ F. We define the following fields associated to F:

• F^{(2)}: the compositum of all cyclic extensions of F of degree p • F^{(3)}: the compositum of all cyclic extensions of F^{(2)} of degree

p which are Galois over F

• F^{(2)}^{(2)}: the compositum of all cyclic extensions of F^{(2)} of degree p.

Then we set W_{F} = Gal(F^{(3)}/F) and V_{F} = Gal(F^{(2)}^{(2)}/F). Observe that for a cyclic extension E/F of degree p, T_{E/F} = Gal(E^{(2)}/F).

Theorem A.3. Let p be prime and F a field with ξ_{p} ∈ F. The following are equivalent:

(1) There exists σ ∈ V_{F} \ Φ(V_{F}) of order p

(2) There exists σ ∈ W_{F} \ Φ(W_{F}) of order p

(3) There exists σ ∈ V_{F} \ Φ(V_{F}) such that for each cyclic E/F of degree p its image σ_{E/F} ∈ T_{E/F} has order at most p.

(4) p = 2 and F is formally real.

If these conditions hold, the elements whose square roots are fixed by σ form an ordering of F.

Proof of Theorem A.3. (1) ⟹ (2) and (1) ⟹ (3). Observe first that F^{(3)} ⊂ F^{(2)}^{(2)}. Hence we have a natural surjection V_{F} ↠ W_{F}. Assume that σ ∈ V_{F} \Φ(V_{F}) has order p. Then its image in W_{F} \Φ(W_{F}) also has order p. Moreover, for any cyclic extension E/F of degree p, the image σ_{E/F} of σ in T_{E/F} has order at most p, as follows. Since E^{(2)} is contained in F^{(2)}^{(2)} we see that σ_{E/F}^{p} = 1 in T_{E/F}.

(2) ⟹ (4). Suppose that σ ∈ W_{F} \ Φ(W_{F}) has order p. As in the proof of Theorem A.2, let a ∈ F^{×} \ F^{×p} be arbitrary such that F (\sqrt[p]{a}) ⊄ F (ξ_{p^{2}}). Set K_{a} := F (\sqrt[p]{a}, ξ_{p^{2}}) ⊂ F^{(2)}. Then [K_{a} : F (ξ_{p^{2}})] = p and L_{a} := F ( \sqrt[p^{2}]{a}, ξ_{p^{2}}) is a cyclic extension of degree p^{2} of F (ξ_{p^{2}}). Moreover, L_{a}/F is a Galois extension of F and L_{a} ⊂ F^{(3)} since Gal(L_{a}/F) is a central extension of degree ≤ p of Gal(K_{a}/F). Now if σ(\sqrt[p]{a}) ≠ \sqrt[p]{a} then σ has order p^{2} in W_{F}. Hence σ fixes all \sqrt[p]{a} for a ∈ F^{×} with F ≠ F (\sqrt[p]{a}) ⊄ F (ξ_{p^{2}}). Since σ ∉ Φ(W_{F}), there must exist a cyclic extension E ⊂ F^{(2)} of F which is not fixed by σ. We deduce that ξ_{p^{2}} ∉ F and so E = F (ξ_{p^{2}}). If p is odd then L := F (ξ_{p^{3}}) is a cyclic extension of F of degree p^{2} and σ restricts to a generator of Gal(L/F) ≃ ℤ/p^{2}ℤ, again a contradiction, whence p = 2. Now let s ∈ W_{F} \ Φ(W_{F}) be an element of order 2. By [MSp1, proof of Theorem 2.7], the elements of F whose square roots are fixed by s form an ordering of F. Hence (2) ⟹ (4).

(4) ⟹ (1). Suppose that (4) holds. By [Be, Satz 3], there exists an ordering of F whose square roots are fixed by some element s of order 2 in G_{F} (2). Then the restriction of s to F^{(2)}^{(2)} is the required element σ ∈ V_{F} \ Φ(V_{F}) of order 2. Hence (4) ⟹ (1).

(3) ⟹ (2). Let σ ∈ V_{F} \ Φ(V_{F}) such that for each cyclic extension E/F of degree p the image σ_{E/F} of σ in T_{E/F} has order at most p. Let E = F (\sqrt[p]{a}) such that σ acts nontrivially on \sqrt[p]{a} and L_{a} = F ( \sqrt[p^{2}]{a}, ξ_{p^{2}}). As above, since the restriction of σ_{E/F} to L_{a} ⊂ E^{(2)} is not of order p^{2}, we deduce that p = 2 and \sqrt{−1} ∉ F. Hence F is not quadratically closed. If F is real closed then there exists precisely one extension E/F of degree 2, namely F^{(2)}, and F^{(2)} = F^{(3)} = F^{(2)}^{(2)}, whence W_{F} = T_{E/F} = V_{F}. Otherwise, F^{(3)} is a compositum of extensions L/F such that Gal(L/F) is either a cyclic group of order 4 or a dihedral group of order 8. (See [MSp2, Corollary 2.18].) On the other hand, each such L lies in E^{(2)} for a suitable quadratic extension E/F: each L may be obtained as a Galois closure of E(\sqrt{γ}) for some [E : F] = 2 and γ ∈ E^{×}. Therefore the restrictions σ_{L/F} of σ to the extensions L/F have order ≤ 2, and so the restriction of σ_{F^{(3)}/F} of σ to F^{(3)}/F has order 2. Hence (3) ⟹ (2). □

3. Pro-p-Groups That are Not Absolute Galois Groups

Theorem A.4. Let p > 3 be prime. There exists a pro-p-ℤ_{p} operator group Ω such that no group of the form Γ := ((Ω ⋆ Σ) ⋊ ℤ_{p})/E, where Σ is any pro-p-group with trivial ℤ_{p}-action, and E is any normal closed subgroup of (Ω⋆Σ)⋊ℤ_{p} such that E ⊂ ((Ω⋆Σ)⋊pℤ_{p})^{(3)}, is an absolute Galois group.

Here ⋆ denotes the free product in the category of pro-p-groups. (Recall that R^{(n)} denotes the nth term of the p-descending series of a pro-p-group R; see section 1.)

The Ω of the theorem is that of the following proposition. Recall that for a pro-p-group Γ, the decomposable subgroup of H^{2}(Γ, 𝔽_{p}) is defined to be the subgroup generated by cup products of elements of H^{1}(Γ, 𝔽_{p}): H^{2}(Γ, 𝔽_{p})^{dec} = H^{1}(Γ, 𝔽_{p}).H^{1}(Γ, 𝔽_{p}). We say that H^{2}(Γ, 𝔽_{p}) is decomposable if H^{2}(Γ, 𝔽_{p}) = H^{2}(Γ, 𝔽_{p})^{dec}.

Proposition A.2. Let p > 3 be prime and C be a cyclic group of order p. There exists a torsion-free pro-p-C operator group Ω such that as 𝔽_{p}C-modules, H^{1}(Ω, 𝔽_{p}) ≃ M_{p} and H^{2}(Ω, 𝔽_{p}) = H^{2}(Ω, 𝔽_{p})^{dec} ≃ M_{p−1} ⊕ \frac{(p−3)}{2}M_{p}.

Here we use M_{i} to denote the unique cyclic 𝔽_{p}C-module of dimension i, and we write the action of 𝔽_{p}C multiplicatively.

Proof. Let D = ⟨g | g^{p} = 1⟩ be a cyclic group of order p, and let ℤ_{p}D be the p-adic group ring, written multiplicatively as G, where the element ḡ_{i} of G corresponds to the element g^{i} of ℤ_{p}D. We interpret the suffixes mod p. Now let C = ⟨σ | σ^{p} = 1⟩ be another group of order p, acting on G via σ(ḡ_{i}) = ḡ_{i−1}. In this way G and H^{1}(G, ℤ_{p}) are free ℤ_{p}C-modules, and H^{1}(G, ℤ_{p}) has a topological generating set y_{0}, y_{1}, …, y_{p−1} dual to ḡ_{0}, ḡ_{1}, …, ḡ_{p−1}. Observe that σ(y_{i}) = y_{i+1}. Next let H = ℤ_{p} be a trivial ℤ_{p}C-module with generator h, and let z ∈ H^{1}(H, ℤ_{p}) be dual to h. We define a nonsplit central extension Ω of H by G as follows. The group H^{2}(G, H) = ⋀^{2} H^{1}(G, H) is a free ℤ_{p}C-module of rank p(p − 1)/2 with free generators y_{0}y_{j} for 1 ≤ j ≤ (p − 1)/2. Consider the element

y := (1 + σ + ⋯ + σ^{p−1})y_{0}y_{1} = y_{0}y_{1} + y_{1}y_{2} + ⋯ + y_{p−1}y_{0}.

Let Ω be the central extension of H by G corresponding to y ∈ H^{2}(G, H).

The group Ω is a torsion-free nilpotent pro-p-group of Hirsch length p + 1. The standard correspondence of group extensions with H^{2} gives us that for suitable representatives g_{i} ∈ Ω with images ḡ_{i} ∈ G, we have the relations [g_{i}, g_{j}] = h for j = i + 1; [g_{i}, g_{j}] = 1 for j ≠ i + 1 and i ≠ j + 1; and [g_{i}, h] = 1 for all i. The action of C on G may be extended to Ω by σ(g_{i}) = g_{i−1}, 0 ≤ i < p, and σ(h) = h.

The E_{2} page of the spectral sequence H^{s}(G, H^{t}(H, 𝔽_{p})) ⇒ H^{s+t}(Ω, 𝔽_{p}) is generated by anticommuting elements ȳ_{i} ∈ E_{2}^{1,0} and z̄ ∈ E_{2}^{0,1}, and we have d_{2}^{0,1}(z̄) = ȳ, where ȳ is the reduction mod p of y. Observe that d_{2}^{0,1} is injective on E^{0,1} = H^{1}(H, 𝔽_{p}). By the five-term exact sequence, H^{1}(Ω, 𝔽_{p}) ≃ H^{1}(G, 𝔽_{p}) as 𝔽_{p}C-modules and so H^{1}(Ω, 𝔽_{p}) ≃ M_{p}.

Now consider H^{2}(Ω, 𝔽_{p}). We claim first that d_{2}^{1,1} is injective. It is enough to show that d_{2}^{1,1}(z̄.x̄) = 0 implies that x̄ = 0 for x̄ ∈ H^{1}(G, 𝔽_{p}). Write x̄ = ∑_{i=0}^{p−1} c_{i}ȳ_{i}, c_{i} ∈ 𝔽_{p}. Since p > 3, the set of elements ȳ_{i}.ȳ_{i+2}.ȳ_{i+3}, 0 ≤ i < p, is 𝔽_{p}-independent and may be expanded to an 𝔽_{p}-basis of E_{2}^{3,0} = H^{3}(G, 𝔽_{p}) consisting of threefold products of ȳ_{i}. Consider the coefficient β_{i} of ȳ_{i}.ȳ_{i+2}.ȳ_{i+3} in an expansion of d_{2}^{1,1}(z̄.x̄). Since p > 3, the only consecutive pair of indices in ȳ_{i}.ȳ_{i+2}.ȳ_{i+3} is {i+2, i+3}. Hence β_{i} depends only on c_{i}. If all β_{i} = 0, then each c_{i} = 0 and therefore d_{2}^{1,1} is injective. Now since d_{2}^{1,1} is injective, E_{3}^{1,1} = E_{∞}^{1,1} = 0, and because E_{2}^{0,2} = 0 we have E_{∞}^{0,2} = 0. Next observe that ȳ = (σ −1)^{p−1}ȳ_{0}.ȳ_{1} and so ȳ ∈ H^{2}(G, 𝔽_{p})^{C}. Since E^{2,0} = H^{2}(G, 𝔽_{p}) is the direct sum of free 𝔽_{p}C-modules M(0, i) ≃ M_{p} on generators ȳ_{0}.ȳ_{i}, 1 ≤ i ≤ (p − 1)/2, we deduce that E_{∞}^{2,0} = E_{3}^{2,0} = (M(0, 1)/M(0, 1)^{C})⊕ \frac{(p−3)}{2}M(0, i). Thus H^{2}(Ω, 𝔽_{p}) ≃ M_{p−1}⊕ \frac{(p−3)}{2}M_{p}. Finally, since H^{1}(Ω, 𝔽_{p}) ≃ H^{1}(G, 𝔽_{p}) and H^{2}(Ω, 𝔽_{p}) is a quotient of E_{2}^{2,0} = H^{2}(G, 𝔽_{p}) = ⋀^{2} H^{1}(G, 𝔽_{p}), H^{2}(Ω, 𝔽_{p}) is decomposable as well. □

Given a free pro-p-group V and a pro-p-group Δ, we say that a surjection V ↠ Δ is a minimal presentation of Δ if inf : H^{1}(Δ, 𝔽_{p}) → H^{1}(V, 𝔽_{p}) is an isomorphism.

Proposition A.3. Let Δ be a pro-p-group, V a free pro-p-group, and 1 → R → V → Δ → 1 a minimal presentation of Δ. Then we have natural maps

(1) H^{2}(Δ, 𝔽_{p}) ≃ (R/(R^{p}[R, V]))^{∨}

(2) H^{2}(Δ, 𝔽_{p})^{dec} ≃ (R/(R ∩ V^{(3)}))^{∨}.

Proof. Set Δ^{[2]} := Δ/Δ^{(2)} and V^{[3]} := V/V^{(3)}. Because the presentation is minimal, Δ^{[2]} ≃ V/V^{(2)}. We then have the following commutative diagram

[figure: 1; R; V; Δ; 1; 1; \frac{V^{(2)}}{V^{(3)}}; V^{[3]}; Δ^{[2]}; 1.]

Passing to the natural maps induced by the Hochschild-Serre-Lyndon spectral sequence with coefficients in 𝔽_{p}, we obtain the commutative diagram

\xymatrix{H^{1}(Δ^{[2]}) \ar[r]^{inf} \ar[d] & H^{1}(V^{[3]}) \ar[r]^{res} \ar[d] & H^{1} (\frac{V^{(2)}}{V^{(3)}} )^{Δ^{[2]}} \ar[r]^{tra} \ar[d] & H^{2}(Δ^{[2]}) \ar[r] \ar[d] & … \\ H^{1}(Δ) \ar[r]^{inf} & H^{1}(V) \ar[r]^{res} & H^{1}(R)^{Δ} \ar[r]^{tra} & H^{2}(Δ) \ar[r] & 0.}

Since the extension 1 → V^{(2)}/V^{(3)} → V^{[3]} → Δ^{(2)} → 1 is central, H^{1}(V^{(2)}/V^{(3)}, 𝔽_{p})^{Δ^{[2]}} = H^{1}(V^{(2)}/V^{(3)}, 𝔽_{p}). Additionally using the fact that the inflation map on the first cohomology group in each row of the previous diagram is an isomorphism, we may extract the following commutative square, with the rightmost transgression map an isomorphism:

\xymatrix{H^{1} (\frac{V^{(2)}}{V^{(3)}} , 𝔽_{p}) \ar[d]^{tra} \ar[r] & H^{1}(R, 𝔽_{p})^{Δ} \ar[d]^{tra}_{≃} \\ H^{2}(Δ^{[2]}, 𝔽_{p}) \ar[r] & H^{2}(Δ, 𝔽_{p})}

Since H^{1}(R, 𝔽_{p})^{Δ} ≃ (R/R^{p}[R, V])^{∨}, we have (1).

The leftmost transgression map, however, is also an isomorphism by [Ho, 1.1 and proof]. Now consider the natural map R/R^{p}[R, V] → V^{(2)}/V^{(3)} of abelian topological groups of exponent p. The image of the natural map H^{1}(V^{(2)}/V^{(3)}, 𝔽_{p}) → H^{1}(R, 𝔽_{p})^{Δ} is H^{1}(RV^{(3)}/V^{(3)}, 𝔽_{p}). We then factor the horizontal maps of the commutative square into homomorphisms followed by injections:

\xymatrix{H^{1} (\frac{V^{(2)}}{V^{(3)}} , 𝔽_{p}) \ar[d]^{tra}_{≃} \ar[r] & H^{1}(\frac{RV^{(3)}}{V^{(3)}} , 𝔽_{p}) \ar@{^{(}->}[r] & H^{1}(R, 𝔽_{p})^{Δ} \ar[d]^{tra}_{≃} \\ H^{2}(Δ^{[2]}, 𝔽_{p}) \ar[r] & H^{2}(Δ, 𝔽_{p})^{dec} \ar@{^{(}->}[r] & H^{2}(Δ, 𝔽_{p}).}

We obtain isomorphisms

H^{1}(R/(R ∩ V^{(3)}), 𝔽_{p}) ≃ H^{1}(RV^{(3)}/V^{(3)}, 𝔽_{p}) ≃ H^{2}(Δ, 𝔽_{p})^{dec}.

Hence H^{2}(Δ, 𝔽_{p})^{dec} ≃ (R/(R ∩ V^{(3)}))^{∨}, and we have proved (2). □

Proposition A.4. Let Γ and H be pro-p-groups with maximal closed subgroups Δ and N, respectively, and α : Γ → H a surjection with α(Δ) = N and ker α ⊂ Δ^{(3)}. Write G for Γ/Δ ≃ H/N. Then as 𝔽_{p}G-modules, H^{2}(Δ, 𝔽_{p})^{dec} ≃ H^{2}(N, 𝔽_{p})^{dec}.

Proof. We prove that H^{2}(Δ, 𝔽_{p})^{dec} ≃ H^{2}(N, 𝔽_{p})^{dec} under a natural isomorphism, and it will follow that the isomorphism is 𝔽_{p}G-equivariant. Let θ : V → Δ be a minimal presentation of Δ with kernel R. We may choose a section W ⊂ V^{(3)} of ker α under the surjection θ. We obtain a minimal presentation of N = α(Δ): 1 → RW → V ⟶^{ψ} N → 1, where ψ = α ∘ θ. By Proposition A.3(2), H^{2}(Δ, 𝔽_{p})^{dec} ≃ (RV^{(3)}/V^{(3)})^{∨} ≃ (RW V^{(3)}/V^{(3)})^{∨} ≃ H^{2}(N, 𝔽_{p})^{dec}. □

Lemma A.5. Let p be prime and Γ a nonfree pro-p-group which is the absolute Galois group of a field F. Then char F ≠ p and ξ_{p} ∈ F.

Proof. Since Γ is not free, then by Witt’s Theorem, char F ≠ p. Since ([F (ξ_{p}) : F], p) = 1, ξ_{p} ∈ F. □

Proposition A.5. Let p > 3 be prime. Suppose that Γ is a prop-group and Δ is a maximal closed subgroup of Γ. If the 𝔽_{p}(Γ/Δ)- module H^{2}(Δ, 𝔽_{p})^{dec} contains a cyclic summand of dimension i with 3 ≤ i < p, then Γ is not an absolute Galois group. Moreover, if Γ contains a normal closed subgroup Λ ⊂ Δ with Γ/Λ ≃ ℤ/p^{2}, then we may take 2 ≤ i < p in the same statement.

Proof. Suppose that Γ = G_{F} for some field F. Then Δ = G_{E} for some E/F of degree p. Since H^{2}(Δ, 𝔽_{p}) ≠ 0, G_{E} is not a free pro-p-group and therefore neither is G_{F} [S, Corollary 3, §I.4.2]. Lemma A.5 gives char F ≠ p and ξ_{p} ∈ F. By [MeSu, Theorem 11.5], H^{2}(Δ, 𝔽_{p}) is decomposable. Therefore by [LMS, Theorem 1], H^{2}(Δ, 𝔽_{p})^{dec} contains no cyclic 𝔽_{p}(Γ/Δ)-summand of dimension i with 3 ≤ i < p, a contradiction. Moreover, if Λ ⊂ N is a normal closed subgroup and Γ/Λ ≃ ℤ/p^{2}ℤ, then by [A], ξ_{p} ∈ N_{E/F} (E). Letting E = F (\sqrt[p]{a}) for some a ∈ F^{×}, we obtain in H^{2}(Γ, 𝔽_{p}) that (a).(ξ_{p}) = 0. By [LMS, Theorem 1], H^{2}(Δ, 𝔽_{p})^{dec} contains no cyclic 𝔽_{p}(Γ/Δ)-summand of dimension 2, again a contradiction. □

Corollary A.5. Let p > 3 be prime. Suppose that Γ and H are pro-p-groups with respective maximal subgroups Δ and N, and α : Γ → H a surjection with α(Δ) = N and ker α ⊂ Δ^{(3)}. Write G for Γ/Δ ≃ H/N. If either H^{2}(Δ, 𝔽_{p})^{dec} or H^{2}(N, 𝔽_{p})^{dec} contains a cyclic 𝔽_{p}G-summand of dimension i with 3 ≤ i < p, then neither Δ nor H is an absolute Galois group.

Moreover, suppose additionally that H contains a normal closed subgroup Λ ⊂ N with H/Λ ≃ ℤ/p^{2}ℤ. Then if either H^{2}(Δ, 𝔽_{p})^{dec} or H^{2}(N, 𝔽_{p})^{dec} contains a cyclic 𝔽_{p}G-summand of dimension 2, then H is not an absolute Galois group.

Proof. By Proposition A.4 we have that H^{2}(Δ, 𝔽_{p})^{dec} ≃ H^{2}(N, 𝔽_{p})^{dec} as 𝔽_{p}G-modules. The remainder follows from Proposition A.5. □

Proof of Theorem A.4. Let Ω be the group of Proposition A.2. Observe that Δ := ((Ω ⋆ Σ) × pℤ_{p})/E is a maximal closed subgroup of Γ of index p. Let G = Γ/Δ and note that the actions of G and C on Δ are identical. By Corollary A.5 it is enough to show that H^{2}(Δ, 𝔽_{p})^{dec} for E = 1 contains a cyclic 𝔽_{p}G-summand M_{i} with 3 ≤ i < p. Assume then that E = 1. By [NSW, Theorem 4.1.4] we have H^{1}(Δ, 𝔽_{p}) ≃ H^{1}(Ω, 𝔽_{p}) ⊕ H^{1}(Σ, 𝔽_{p}) ⊕ H^{1}(pℤ_{p}, 𝔽_{p}) and H^{2}(Δ, 𝔽_{p}) ≃ H^{2}(Ω, 𝔽_{p}) ⊕ H^{2}(Σ, 𝔽_{p}) ⊕ H^{1}(Ω, 𝔽_{p}) ⊕ H^{1}(Σ, 𝔽_{p}). By Proposition A.2, H^{2}(Ω, 𝔽_{p}) is decomposable and so H^{2}(Ω, 𝔽_{p}) is a direct summand of H^{2}(Δ, 𝔽_{p})^{dec}. From Proposition A.2, we obtain that H^{2}(Ω, 𝔽_{p}) contains an 𝔽_{p}G-summand M_{i} with 3 ≤ i ≤ p − 1. □

Remark. The proof excludes the groups Γ from the class of absolute Galois groups by using the n = 2 case of [LMS, Theorem 1]. However, neither a direct application of the n = 1 case nor Theorem A.2 excludes the groups Γ. Observe also that the fact that Ω ⋊ ℤ_{p} is not an absolute Galois group could be deduced from the main results in [Koe], using different methods. However, the fact that each ((Ω × Σ) ⋊ ℤ_{p})/E is not an absolute Galois group does not follow from [Koe].

Acknowledgement

We thank the referee of [LMS] for suggesting that we illustrate [LMS, Theorem 1] by constructing examples of pro-p-groups which are not absolute Galois groups, for this appendix is the result. The third author would also like to thank Sunil Chebolu, Ajneet Dhillon and Vahid Shirbisheh for stimulating discussions related to this paper.

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Department of Mathematical Sciences, University of Aberdeen, Meston Building, King’s College, Aberdeen AB24 3UE, Scotland UK

E-mail address: bensondj@maths.abdn.ac.uk

Department of Mathematics, Middlesex College, University of Western Ontario, London, Ontario N6A 5B7 CANADA

E-mail address: nlemire@uwo.ca

E-mail address: minac@uwo.ca

Department of Mathematics, Davidson College, Box 7046, Davidson, North Carolina 28035-7046 USA

E-mail address: joswallow@davidson.edu