GALOIS MODULE STRUCTURE OF GALOIS COHOMOLOGY

NICOLE LEMIRE^{⋆}, JÁN MINÁČ^{∗†}, AND JOHN SWALLOW^{‡}

Abstract. Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group G_{F} of F. We determine the structure of the cohomology group H^{n}(U, 𝔽_{p}) as an 𝔽_{p}[G_{F} /U]-module for all n ∈ ℕ. Previously this structure was known only for n = 1, and until recently the structure even of H^{1}(U, 𝔽_{p}) was determined only for F a local field, a case settled by Borevič and Faddeev in the 1960s.

Let F be a field containing a primitive pth root of unity ξ_{p}. Let G_{F} be the absolute Galois group of F, U an open normal subgroup of G_{F} of index p, and G = G_{F} /U. Let E be the fixed field of U in the separable closure F_{sep} of F. Fix a ∈ F such that E = F (\sqrt[p]{a}), and let σ ∈ G satisfy \sqrt[p]{a}^{σ−1} = ξ_{p}.

In the 1960s Z. I. Borevič and D. K. Faddeev classified the possible G-module structures of the first cohomology groups H^{1}(U, 𝔽_{p}) in the case F a local field [B]. Quite recently this result was extended for all fields F as above [MS]. For the study of Galois cohomology it is important to extend these results to all cohomology groups H^{n}(U, 𝔽_{p}), n ∈ ℕ, and a solution of this problem was out of reach until now.

Recently, based on earlier work of A. S. Merkurjev, M. Rost and A. A. Suslin, V. Voevodsky established the Bloch-Kato Conjecture [V1, V2], and it turns out that some of the main theorems in his proof are sufficient to determine the structure of all G-Galois modules H^{n}(U, 𝔽_{p}), using only simple arithmetical invariants attached to the field extension E/F. The theorems we use (quoted as Theorems 3 and 4 in section 1 below) had, in fact, been standard conjectures on Galois cohomology.

Date: October 29, 2004.

⋆Research supported in part by NSERC grant R3276A01.

∗Research supported in part by NSERC grant R0370A01.

†Supported by the Mathematical Sciences Research Institute, Berkeley.

‡Research supported in part by National Security Agency grant MDA904-02-1-0061.

It is interesting to point out, however, that the case n = 2 could have already been settled some 20 years ago, thanks to the work of Merkurjev and Suslin [MeSu].

The main ingredient for our determination of the G-module structure of H^{n}(U, 𝔽_{p}) is Milnor K-theory. (See [Mi] and [FV, Chap. IX].) For i ≥ 0, let K_{i}F denote the ith Milnor K-group of the field F, with standard generators denoted by {f_{1}, …, f_{i}}, f_{1}, …, f_{i} ∈ F \ {0}. For α ∈ K_{i}F, we denote by ᾱ the class of α modulo p, and we use the usual abbreviation k_{n}F for K_{n}F/pK_{n}F. We write N_{E/F} for the norm map K_{n}E → K_{n}F, and we use the same notation for the induced map modulo p. We denote by i_{E} the natural homomorphism in the reverse direction. We also apply the same notation N_{E/F} and i_{E} for the corresponding homomorphisms between cohomology groups. The image of an element α ∈ K_{i}F in H^{i}(G_{F} , 𝔽_{p}) we also denote by α. Voevodsky’s proof of the Bloch-Kato Conjecture establishes a G_{F} -isomorphism H^{n}(U, 𝔽_{p}) ≅ k_{n}E. We formulate our results in terms of Galois cohomology for intended applications, but we use Milnor K-theory in our proof.

We concentrate upon the case when F is a perfect field, and at the end of the paper we indicate how one may reduce the case of an imperfect field F to the case of characteristic 0.

Our decomposition depends on four arithmetic invariants Υ_{1}, Υ_{2}, y, z, which we define as follows. First, for an element ᾱ of k_{i}F, let

ann_{k_{n−1}F} ᾱ = ann (k_{n−1}F ⟶^{ᾱ·−} k_{n−1+i}F )

denote the annihilator of the product with ᾱ. When the domain of ᾱ is clear, we omit the subscript on the map and write simply ann ᾱ. Because we will often use the elements \overline{\{a\}}, \overline{\{ξ_{p}\}}, \overline{\{a, a\}}, and \overline{\{a, ξ_{p}\}}, we omit the bars for these elements. We also omit the bar in the element \overline{\{\sqrt[p]{a}\}} ∈ k_{n}E.

Fix n ∈ ℕ and U an open normal subgroup of G_{F} of index p with fixed field E. Define invariants associated to E/F and n as follows:

d := dim_{𝔽_{p}} k_{n}F / N_{E/F} k_{n}E
e := dim_{𝔽_{p}} N_{E/F} k_{n}E
Υ_{1} := dim_{𝔽_{p}} ann{a, ξ_{p}} / ann{a}
Υ_{2} := dim_{𝔽_{p}} k_{n−1}F / ann{a, ξ_{p}}

y := \begin{cases}dim_{𝔽_{p}} (N_{E/F} k_{n}E) / \{a\} · k_{n−1}F, & p > 2 \\ dim_{𝔽_{2}} (N_{E/F} k_{n}E) / \{a\} · ann_{k_{n−1}F} \{a, −1\}, & p = 2 \end{cases}

z := \begin{cases}dim_{𝔽_{p}} (k_{n}F) / (\{ξ_{p}\} · W + N_{E/F} k_{n}E) , & p > 2 \\ dim_{𝔽_{2}} (k_{n}F) / (\{a\} · k_{n−1}F + N_{E/F} k_{n}E) , & p = 2, \end{cases}

where W is a complement of ann{a, ξ_{p}} in k_{n−1}F. (In Lemma 1 we show that z is independent of the choice of W.)

Our main results are then the following.

Theorem 1. If p > 2, F is perfect and n ∈ ℕ then

H^{n}(U, 𝔽_{p}) ≅ X_{1} ⊕ X_{2} ⊕ Y ⊕ Z

where

(1) X_{1} is a trivial 𝔽_{p}[G]-module of dimension Υ_{1}

(2) X_{2} is a direct sum of Υ_{2} cyclic 𝔽_{p}[G]-modules of dimension 2 (3) Y is a free 𝔽_{p}[G]-module of rank y

(4) Z is a trivial 𝔽_{p}[G]-module of dimension z.

Further we have

(5) Y^{G} = i_{E}N_{E/F} H^{n}(U, 𝔽_{p})

(6) N_{E/F} : X_{1} ⊕ X_{2} → {a} · H^{n−1}(G_{F} , 𝔽_{p}) is surjective

(7) Υ_{1} + Υ_{2} + y = e

(8) Υ_{2} + z = d

Theorem 2. If p = 2 and n ∈ ℕ then

H^{n}(U, 𝔽_{2}) ≅ X_{1} ⊕ Y ⊕ Z

where

(1) X_{1} is a trivial 𝔽_{2}[G]-module of dimension Υ_{1}

(2) Y is a free 𝔽_{2}[G]-module of rank y.

(3) Z is a trivial 𝔽_{2}[G]-module of dimension z.

Further we have

(4) Y^{G} = i_{E}N_{E/F} H^{n}(U, 𝔽_{2})

(5) N_{E/F} : X_{1} → {a} · ann{a, −1} is an isomorphism

(6) Υ_{1} + y = e

(7) Υ_{2} + z = d Remark. As explained at the end of section 5, if n ≤ 2 in Theorem 1, we may remove the hypothesis that F is perfect. The case n = 1 in the two theorems recovers the results of [MS].

1. Bloch-Kato and Milnor K-theory

Our proof relies on the following two results in Voevodsky’s proof of the Bloch-Kato Conjecture. Because we apply Voevodsky’s results in the case when the base field contains a primitive pth root of unity we shall formulate Voevodsky’s results restricted to this case. The first is the Bloch-Kato Conjecture itself:

Theorem 3 ([V1, Def. 5.1] and [V2, Thm. 7.1]).

(1) Let F be a field containing a primitive pth root of unity and m ∈ ℕ. Then the norm residue homomorphism

k_{m}F → H^{m}(G_{F} , μ_{p})

is an isomorphism.

(2) For any cyclic extension E/F of degree p, the sequence

K_{m}E ⟶^{σ−1} K_{m}E ⟶^{N_{E/F}} K_{m}F

is exact.

The second result establishes an exact sequence connecting k_{m}F and k_{m}E for consecutive m. (We translate the statement of the original result to K-theory using the previous theorem.) In the following result a is chosen to satisfy E = F (\sqrt[p]{a}).

Theorem 4 ([V1, Prop. 5.2]). Let F be a field containing a primitive pth root of unity with no extensions of degree prime to p. Then for any cyclic extension E/F of degree p and m ≥ 1, the sequence

k_{m−1}E ⟶^{N_{E/F}} k_{m−1}F ⟶^{{a}·−} k_{m}F ⟶^{i_{E}} k_{m}E

is exact.

Now if F is a perfect field, we observe that we may remove the hypothesis that the field F has no extensions of degree prime to p. It is precisely to ensure that the sequence above is exact that we require the hypothesis that F is perfect in Theorem 1. We give a proof of the following theorem in section 5.

Theorem 5 (Modification of Theorem 4). Let F be a field containing a primitive pth root of unity, and if p > 2 assume that F is perfect. Then for any cyclic extension E/F of degree p and m ≥ 1 the sequence

k_{m−1}E ⟶^{N_{E/F}} k_{m−1}F ⟶^{{a}·−} k_{m}F ⟶^{i_{E}} k_{m}E

is exact.

2. Notation and Lemmas

For the remainder of the paper, except for section 6, we assume that F is perfect if p > 2. We fix n ∈ ℕ and the cyclic extension E = F (\sqrt[p]{a}), and we write k_{n−1}F = ann{a}⊕V ⊕W, where ann{a, ξ_{p}} = ann{a}⊕V. Observe that Υ_{1} = dim_{𝔽_{p}} V and Υ_{2} = dim_{𝔽_{p}} W. We show that the invariant z is independent of the choice of W:

Lemma 1. The invariant z is independent of the particular complement W of ann{a, ξ_{p}} in k_{n−1}F.

Proof. This is obvious for p = 2, so assume that p > 2. Let W′ be another 𝔽_{p}-subspace of k_{n−1}F with k_{n−1}F = ann{a, ξ_{p}}⊕W′. Then we may choose bases {w_{i}}_{i∈I} and {w′_{i}}_{i∈I} of W and W′, respectively, such that w′_{i} = w_{i}+u_{i} for u_{i} ∈ ann{a, ξ_{p}}. Then {ξ_{p}}·w′_{i} = {ξ_{p}}·w_{i}+{ξ_{p}}·u_{i}, and {ξ_{p}} · u_{i} ∈ ann{a}. Because ann{a} = N_{E/F} k_{n}E by Theorem 5, we see that

{ξ_{p}} · W + N_{E/F} k_{n}E = {ξ_{p}} · W′ + N_{E/F} k_{n}E

and therefore z does not depend on the choice of W. □

We denote by i_{E}: K_{n}F → K_{n}E the map induced by the inclusion of F in E. In what follows we will frequently refer to the element \sqrt[p]{a}, and so we abbreviate it by A. We will also often use the observation that if p = 2 then {a, ξ_{p}} = {a, −1} = {a, a} ∈ k_{2}F, while if p > 2 then {a, a} = 0 ∈ k_{2}F. Finally, we will use the projection formula for taking the norms of standard generators of K_{i}F (see [FW, p. 81]).

Lemma 2. We have the vector space isomorphism

V ⊕ W ⟶^{{a}·−} {a} · k_{n−1}F

and, if p > 2, the compositum of the maps {ξ_{p}} · − and i_{E}

W ⟶^{{ξ_{p}}·−} {ξ_{p}} · W ⟶^{i_{E}} i_{E}({ξ_{p}} · W)

is a vector space isomorphism as well.

Proof. The first isomorphism follows from the fact that V ⊕ W is a complement in k_{n−1}F of the kernel of multiplication by {a}. For the second, assume p > 2. Suppose that w̄ ∈ W and ᾱ = {ξ_{p}} · w̄ ∈ ker i_{E}. Then by Theorem 5, ᾱ = {a} · c̄ for c ∈ K_{n−1}F. Since {a, a} = 0 we see that {a} · ᾱ = 0. But then w̄ ∈ ann{a, ξ_{p}} and so w̄ = 0. □

For γ ∈ K_{n}E, let l(γ) denote the dimension of the cyclic 𝔽_{p}[G]- submodule ⟨γ̄⟩ of k_{n}E generated by γ̄. Then we have

(σ − 1)^{l(γ)−1}⟨γ̄⟩ = ⟨γ̄⟩^{G} ≠ 0 and (σ − 1)^{l(γ)}⟨γ̄⟩ = 0.

We denote by N the map (σ − 1)^{p−1} on k_{n}E. Because (σ − 1)^{p−1} = 1 + σ + ⋯ + σ^{p−1} in 𝔽_{p}[G], we may use i_{E}N_{E/F} and N interchangeably on k_{n}E.

Lemma 3. Suppose p > 2 and γ ∈ K_{n}E.

(1) If 3 ≤ l(γ) ≤ p, then there exists α ∈ K_{n}E such that

⟨Nᾱ⟩ = ⟨γ̄⟩^{G}.

(2) If l(γ) = 2 and

γ̄ ∉ {A} · i_{E}(k_{n−1}F) + (k_{n}E)^{G}

then there exist α ∈ K_{n}E and b ∈ K_{n−1}F such that

⟨Nᾱ⟩ = ⟨γ̄ + {A} · i_{E}(b̄)⟩^{G}.

Proof. Let l = l(γ) and suppose 3 ≤ l ≤ i ≤ p. We show by induction on i that there exists α_{i} ∈ K_{n}E such that ⟨(σ − 1)^{i−1}ᾱ_{i}⟩ = ⟨γ̄⟩^{G}. Then setting α := α_{p}, the proof will be complete. If i = l then α_{i} = γ suffices. Assume now that l ≤ i < p and that our statement is true for i.

Set c = N_{E/F} α_{i}. Since i_{E}c̄ = Nᾱ_{i} = (σ − 1)^{p−1}ᾱ_{i} and i < p, i_{E}c̄ = 0. By Theorem 5, c̄ = {a}·b̄ for b ∈ K_{n−1}F. Equivalently, c = {a}·b+pf for f ∈ K_{n}F. Then

N_{E/F} (α_{i} − ({A} · i_{E}(b) + i_{E}(f))) = 0.

By Theorem 3, there exists ω ∈ K_{n}E such that

(σ − 1)ω = α_{i} − ({A} · i_{E}(b) + i_{E}(f)).

Then (σ−1)^{2}ω = (σ−1)α_{i}−{ξ_{p}}·i_{E}(b). Since i ≥ 3, ⟨(σ − 1)^{i}ω̄⟩ = ⟨γ̄⟩^{G} and we can set α_{i+1} = ω.

For the second part, suppose l = 2 = i. Proceeding in the same way as above, we see that for α_{2} = γ we have N_{E/F} α_{2} = {a} · b + pf for b ∈ K_{n−1}F and f ∈ K_{n}F. As before, there exists ω ∈ K_{n}E such that (σ − 1)ω = α_{2} − ({A} · i_{E}(b) + i_{E}(f)). Then

(σ − 1)^{2}ω = (σ − 1)(α_{2} − {A} · i_{E}(b)) = (σ − 1)(γ − {A} · i_{E}(b)).

Observe that γ̄ − {A} · i_{E}(b̄) ∉ (k_{n}E)^{G} by hypothesis. Therefore l(γ − {A} · i_{E}(b)) = 2 and we can set α_{3} := ω. We may then continue by induction on i as above, concluding that there exists an element α = α_{p} ∈ K_{n}E such that ⟨Nᾱ_{p}⟩ = ⟨(σ − 1)^{p−1}ᾱ_{p}⟩ = ⟨γ̄ − {A} · i_{E}(b̄)⟩^{G}, as required. □

In the following lemma we elongate the exact sequence of Theorem 5.

Lemma 4. The following sequence is exact:

0 → ann{a} → k_{n−1}F ⟶^{{a}·−} k_{n}F ⟶^{i_{E}} (k_{n}E)^{G} ⟶^{N_{E/F}} {a}·ann{a, ξ_{p}} → 0.

Here the map ann{a} → k_{n−1}F is the natural inclusion.

Proof. We show first that N_{E/F} ((k_{n}E)^{G}) ⊂ {a} · ann{a, ξ_{p}}. Let ᾱ ∈ (k_{n}E)^{G} and β = N_{E/F} α. Since i_{E}(N_{E/F} ᾱ) = (σ − 1)^{p−1}ᾱ = 0 we have that β̄ = N_{E/F} ᾱ = {a} · b̄ for some b ∈ K_{n−1}F by Theorem 5.

Suppose p = 2. Since β̄ is in the image of N_{E/F}, we have by Theorem 5 that {a} · β̄ = {a, a} · b̄ = 0. Since {a, a} = {a, −1}, we have b̄ ∈ ann{a, −1}.

Now suppose that p > 2. Write β = {a} · b + pf for some f ∈ K_{n}F. Then by the projection formula

N_{E/F} (α − ({A} · i_{E}(b) + i_{E}(f))) = 0.

By Theorem 3, there exists ω ∈ K_{n}E such that

(σ − 1)ω = α − ({A} · i_{E}(b) − i_{E}(f)).

Then (σ − 1)^{2}ω̄ = {ξ_{p}} · i_{E}(b̄).

If (σ − 1)^{2}ω̄ = 0 then since by Theorem 5, ker i_{E} = {a} · k_{n−1}F,

{ξ_{p}} · b̄ = {a} · h̄

for some h ∈ K_{n−1}F. Because {a, a} = 0, the right-hand side of the preceding equation is annihilated by {a}. Therefore b̄ ∈ ann{a, ξ_{p}}.

If (σ − 1)^{2}ω̄ ≠ 0 then l(ω) = 3 and Lemma 3 shows that

i_{E}(\{ξ_{p}\} · b̄) = cNλ̄ = i_{E}(N_{E/F} (\overline{cλ}))

for some λ ∈ K_{n}E and c ∈ ℤ. Since by Theorem 5, ker i_{E} = {a}·k_{n−1}F we have

\{ξ_{p}\} · b̄ = N_{E/F} (\overline{cλ}) + \{a\} · h̄

for some h ∈ K_{n−1}F. Now by Theorem 5 and the fact that {a, a} = 0, the right-hand side of the preceding equation is annihilated by {a}. Then b̄ ∈ ann{a, ξ_{p}}. Hence in all cases N_{E/F} ᾱ ∈ {a} · ann{a, ξ_{p}}.

Exactness at the first two terms is obvious, and exactness at the third term follows from Theorem 5.

For exactness at the fourth term, suppose

γ̄ ∈ (k_{n}E)^{G} and N_{E/F} γ̄ = 0.

Then N_{E/F} γ = pf for f ∈ K_{n}F. Let β = γ − i_{E}(f). Then N_{E/F} β = 0 and by Theorem 3 there exists α ∈ K_{n}E such that (σ − 1)α = β. If p = 2 then β̄ = i_{E}(N_{E/F} ᾱ) ∈ i_{E}k_{n}F and we are done. Thus assume p > 2.

Now suppose ᾱ ∈ {A} · i_{E}(k_{n−1}F) + (k_{n}E)^{G}. Then

β̄ = (σ − 1)ᾱ ∈ {ξ_{p}} · i_{E}(k_{n−1}F) ⊂ i_{E}(k_{n}F),

and hence γ̄ = β̄ + i_{E}(f̄) ∈ i_{E}(k_{n}F) as well. Otherwise ᾱ ∉ {A} · i_{E}(k_{n−1}F) + (k_{n}E)^{G}. Now if (σ − 1)ᾱ = β̄ = 0 we are done as then γ̄ = i_{E}(f̄). Hence assume (σ−1)ᾱ ≠ 0. Then l(α) = 2 and by Lemma 3 we see that there exist δ ∈ K_{n}E, b ∈ K_{n−1}F and c ∈ ℤ such that

cNδ̄ = (σ − 1)(ᾱ + {A} · i_{E}(b̄)) = β̄ + {ξ_{p}} · i_{E}(b̄).

Thus β̄ = cNδ̄ − {ξ_{p}} · i_{E}(b̄) ∈ i_{E}(k_{n}F) and exactness at the fourth term is established.

Finally we show the exactness at the fifth term. Since

{a} · ann{a, ξ_{p}} = {a} · V

it is enough to show that each element {a} · v̄ where v̄ ∈ V can be written as N_{E/F} ᾱ for some ᾱ ∈ (k_{n}E)^{G}. Observe that (σ − 1)({A} · i_{E}v̄) = {ξ_{p}} · i_{E}(v̄). Also we have

N_{E/F} (\{A\} · i_{E}(v̄)) = \begin{cases}\{a\} · v̄ & if p > 2 \\ \{−a\} · v̄ & if p = 2. \end{cases}

Therefore it is enough to show that there exists an element γ̄ ∈ k_{n}E such that (σ − 1)γ̄ = {ξ_{p}} · i_{E}(v̄) and

N_{E/F} γ̄ = \begin{cases}0 & if p > 2 \\ \{−1\} · v̄ & if p = 2. \end{cases}

Indeed then we can set ᾱ = {A} · i_{E}(v̄) − γ̄.

Because v̄ ∈ ann{a, ξ_{p}} we see that {ξ_{p}} · i_{E}(v̄) ∈ ann{a}. By Theorem 5 there exists β̄ ∈ k_{n}E such that

{ξ_{p}} · v̄ = N_{E/F} β̄ and i_{E}(N_{E/F} β̄) = (σ − 1)^{p−1}β̄.

Then setting γ̄ = (σ − 1)^{p−2}β̄ we obtain our required element. The proof of our lemma has now been completed. □

Finally, we need a general lemma about 𝔽_{p}[G]-modules.

Lemma 5 (Exclusion Lemma). Let M_{1} and M_{2} be 𝔽_{p}[G]-modules contained in a common 𝔽_{p}[G]-module. Suppose that M_{1}^{G} ∩ M_{2}^{G} = {0}. Then M_{1} + M_{2} = M_{1} ⊕ M_{2}.

Proof. Let M = M_{1} ∩ M_{2} and suppose that m ∈ M \ {0}. Let

m̃ = (σ − 1)^{l(m)−1}(m) ≠ 0.

Then m̃ ∈ M_{1}^{G} ∩ M_{2}^{G}, a contradiction. Hence M_{1} ∩ M_{2} = {0} and M_{1} + M_{2} = M_{1} ⊕ M_{2}. □

3. Construction of Submodules

Proposition 1. k_{n}E contains a submodule X_{1} such that

• X_{1} is a trivial 𝔽_{p}[G]-module of dimension Υ_{1}

• X_{1} ∩ i_{E}k_{n}F = {0}

• N_{E/F} restricts to an isomorphism X_{1} → {a} · V.

Moreover, if p > 2, then k_{n}E contains a submodule X_{2}, independent of X_{1}, such that

• X_{2} is a direct sum of Υ_{2} cyclic submodules of dimension 2 and dim_{𝔽_{p}} X_{2}^{G} = Υ_{2}.

• (X_{1} + X_{2}) ∩ i_{E}k_{n}F = (σ − 1)X_{2} = X_{2}^{G} = i_{E}({ξ_{p}} · W)

• We have an exact sequence

0 → {ξ_{p}} · W ⟶^{i_{E}} X_{1} + X_{2} ⟶^{N_{E/F}} {a} · k_{n−1}F → 0

Proof. Let I be an 𝔽_{p}-basis for V. Let v̄ be an arbitrary element of I, and consider ᾱ = {A} · i_{E}v̄. Now (σ − 1)ᾱ = i_{E}({ξ_{p}} · v̄).

Since v̄ ∈ ann{a, ξ_{p}} we see that {ξ_{p}}·v̄ ∈ ann_{k_{n}F} {a}. By Theorem 5

{ξ_{p}} · v̄ = N_{E/F} β̄ and i_{E}(N_{E/F} β̄) = Nβ̄ = (σ − 1)^{p−1}β̄

for some β ∈ K_{n}E. Set γ = (σ − 1)^{p−2}β and x̄_{v} = ᾱ − γ̄ ∈ k_{n}E.

If p = 2 then

N_{E/F} x̄_{v} = {−a} · v̄ − N_{E/F} γ̄ = {−a} · v̄ − {−1} · v̄ = {a} · v̄.

If p > 2, then observe that since γ is in the image of σ − 1 we have \overline{N_{E/F} γ} = 0. Then, by the projection formula

N_{E/F} x̄_{v} = \{a\} · v̄ − \overline{N_{E/F} γ} = \{a\} · v̄.

Now in either case, since (σ − 1)^{p−1}β̄ = i_{E}(N_{E/F} β̄),

(σ − 1)x̄_{v} = i_{E}({ξ_{p}} · v̄) − (σ − 1)^{p−1}β̄ = i_{E}({ξ_{p}} · v̄) − i_{E}({ξ_{p}} · v̄) = 0.

Set

X_{1} := ⊕_{v̄∈I}⟨x̄_{v}⟩.

We have shown that X_{1} is a trivial 𝔽_{p}[G]-module. Moreover, because N_{E/F} x̄_{v} = {a} · v̄ and {a} · − is injective on V by Lemma 2,

N_{E/F} |_{X_{1}} : X_{1} → {a} · V

takes a basis of X_{1} to a basis of {a} · V and dim_{𝔽_{p}} X_{1} = dim_{𝔽_{p}} V = Υ_{1}. Finally, since N_{E/F} is trivial on i_{E}k_{n}F, we have X_{1} ∩ i_{E}k_{n}F = {0}.

Now suppose that p > 2. Set

X_{2} := ({A} · i_{E}W) + i_{E}({ξ_{p}} · W).

Let w̄ ∈ W and consider x̄_{w} = {A} · i_{E}(w̄).

Since (σ − 1)x̄_{w} = i_{E}({ξ_{p}} · w̄) and (σ − 1)({ξ_{p}} · w̄) = 0, we obtain (σ−1)X_{2} = i_{E}({ξ_{p}}·W). Hence on {A}·i_{E}W, σ−1 acts as i_{E}({ξ_{p}}·−), which by Lemma 2 is an isomorphism of vector spaces. Hence σ − 1 is an isomorphism as well. Moreover, if an arbitrary {A} · i_{E}(w̄_{1}) + {ξ_{p}} · i_{E}(w̄_{2}) ∈ X_{2} lies in the kernel of σ − 1, w̄_{1} = 0. Hence X_{2}^{G} = i_{E}({ξ_{p}} · W). Since we already observed that X_{1} ∩ i_{E}k_{n}F = {0} we see that X_{2}^{G} ∩X_{1} = {0} and by Lemma 5 we conclude that X_{1} +X_{2} = X_{1} ⊕X_{2}.

By the projection formula N_{E/F} x̄_{w} = {a} · w̄ and by the definition of W, {a} · w̄ = 0 implies w̄ = 0. Since N_{E/F} ({ξ_{p}} · i_{E}(w̄_{2})) = 0 for all w̄_{2} ∈ W, we deduce that restricted to X_{2}, N_{E/F} surjects X_{2} onto {a}·W with kernel i_{E}({ξ_{p}} · W). By Lemma 2, {a} · k_{n−1}F = {a} · (V + W); hence on X_{1} ⊕ X_{2}, N_{E/F} is a surjection onto {a} · k_{n−1}F with kernel i_{E}({ξ_{p}} · W).

Finally observe that N_{E/F} i_{E}k_{n}F = {0}. Hence

(X_{1} + X_{2}) ∩ i_{E}k_{n}F ⊂ i_{E}({ξ_{p}} · W).

Since i_{E}({ξ_{p}} · W) ⊂ i_{E}k_{n}F, we have equality.

Now we have shown that σ − 1 is an isomorphism of vector spaces {A} · i_{E}W → i_{E}({ξ_{p}} · W), and by Lemma 2, we have an isomorphism W → i_{E}({ξ_{p}} · W). Therefore X_{2} is a direct sum of cyclic submodules ⟨x̄_{w}⟩ of dimension 2, with x̄_{w} in one-to-one correspondence with basis elements of W. Hence the direct sum contains Υ_{2} cyclic summands. □

If p = 2, let X = X_{1} be a submodule of k_{n}E satisfying the conditions of the preceding proposition. If p > 2, let X = X_{1} + X_{2} for X_{1}, X_{2} satisfying the conditions of the same.

Proposition 2. k_{n}E contains a submodule Y independent from X such that

• Y is a free 𝔽_{p}[G]-module of rank

y = \begin{cases}dim_{𝔽_{p}}(N_{E/F} k_{n}E)/\{a\} · k_{n−1}F, & p > 2 \\ dim_{𝔽_{2}}(N_{E/F} k_{n}E)/\{a\} · ann_{k_{n−1}F} \{a, −1\}, & p = 2 \end{cases}

• Y^{G} = i_{E}N_{E/F} k_{n}E

• if p > 2, Υ_{1} + Υ_{2} + y = e

• if p = 2, Υ_{1} + y = e

Proof. Let I be a basis for the subspace i_{E}(N_{E/F} k_{n}E). For each basis element ȳ ∈ I, let α_{y} ∈ K_{n}E satisfy i_{E}(N_{E/F} ᾱ_{y}) = ȳ. Then ⟨ᾱ_{y}⟩ is a cyclic submodule of dimension p, hence isomorphic to 𝔽_{p}[G], with

⟨ᾱ_{y}⟩^{G} = (σ − 1)^{p−1}⟨ᾱ_{y}⟩ = ⟨Nᾱ_{y}⟩ = ⟨ȳ⟩.

Set

Y = ∑_{ȳ∈I}⟨ᾱ_{y}⟩.

By Lemma 5, Y = ⊕_{ȳ∈I}⟨ᾱ_{y}⟩ and so Y is a free 𝔽_{p}[G]-module. Moreover,

Y^{G} = (σ − 1)^{p−1}Y = NY = ⊕_{ȳ∈I}⟨ȳ⟩ = i_{E}(N_{E/F} k_{n}E).

Now the rank of Y is equal to the dimension of i_{E}(N_{E/F} k_{n}E), or

dim_{𝔽_{p}}(N_{E/F} k_{n}E)/ ((N_{E/F} k_{n}E) ∩ ker i_{E}) .

Now by Theorem 5, N_{E/F} (k_{n}E) = ann{a}, and by the same Theorem, ker i_{E} = {a} · k_{n−1}F. Hence

N_{E/F} (k_{n}E) ∩ ker i_{E} = ann_{k_{n}F} {a} ∩ {a} · k_{n−1}F.

Suppose that p = 2. Since {a, a} = {a, −1} we deduce that

N_{E/F} (k_{n}E) ∩ ker i_{E} = {a} · ann{a, −1}.

The dimension of this subspace is equal to dim_{𝔽_{p}} ann{a, −1}/ ann{a}, or Υ_{1}.

Now suppose that p > 2. Since {a, a} = 0, {a} · k_{n−1}F ⊂ ann{a} and we deduce that

N_{E/F} (k_{n}E) ∩ ker i_{E} = {a} · k_{n−1}F,

which is of dimension dim_{𝔽_{p}}(k_{n−1}F)/ ann{a}, or Υ_{1} + Υ_{2}.

As e = dim_{𝔽_{p}} N_{E/F} k_{n}E, we deduce that if p = 2 then Υ_{1}+rank Y = e and if p > 2, Υ_{1} + Υ_{2} + rank Y = e.

Now we claim that Y is independent from X. Suppose β̄ ∈ X^{G} ∩Y^{G}. Now Y^{G} = i_{E}(N_{E/F} k_{n}E) ⊂ i_{E}(k_{n}F), so β̄ = i_{E}(ᾱ) where ᾱ = N_{E/F} γ̄ for γ ∈ K_{n}E. If p = 2 then by Proposition 1, i_{E}(k_{n}F) ∩ X = {0}, and so X ∩ Y = {0} by Lemma 5.

If p > 2, Proposition 1 tells us that

i_{E}(k_{n}F) ∩ X = X_{2}^{G} = {ξ_{p}} · i_{E}W.

Hence β̄ = {ξ_{p}} · i_{E}(w̄) for w̄ ∈ W. Since i_{E}({ξ_{p}} · w̄) = i_{E}(ᾱ) and by Theorem 5, ker i_{E} = {a} · k_{n−1}F,

{ξ_{p}} · w̄ = ᾱ + ({a} · f̄)  (1)

for f ∈ K_{n−1}F. Now because ᾱ ∈ N_{E/F} k_{n}E, by Theorem 5, {a}·ᾱ = 0. Moreover, {a, a} = 0 since we have assumed that p > 2. Hence the right-hand side of (1) is annihilated by multiplication by {a}. Therefore w̄ ∈ ann{a, ξ_{p}}, and by the definition of W, w̄ = 0. By Lemma 5, X + Y = X ⊕ Y. □

Now let X and Y be submodules satisfying the conditions of the preceding propositions.

Proposition 3. k_{n}E contains a submodule Z independent from X +Y such that

• Z is a trivial 𝔽_{p}[G]-module of dimension

z = \begin{cases}dim_{𝔽_{p}}(k_{n}F)/(\{ξ_{p}\} · W + N_{E/F} k_{n}E), & p > 2 \\ dim_{𝔽_{2}}(k_{n}F)/(\{a\} · k_{n−1}F + N_{E/F} k_{n}E), & p = 2 \end{cases}

• (k_{n}E)^{G} = X^{G} + Y^{G} + Z

• Υ_{2} + z = d

Proof. Let Z be a complement of (X^{G} +Y^{G})∩i_{E}(k_{n}F) in i_{E}(k_{n}F). By Lemma 5, (X + Y) + Z = (X + Y) ⊕ Z.

Clearly X^{G} + Y^{G} + Z ⊂ (k_{n}E)^{G}. Now suppose ᾱ ∈ (k_{n}E)^{G} and let β = N_{E/F} α. By Lemma 4, β̄ = {a} · b̄ for some b̄ ∈ ann{a, ξ_{p}}.

Let v̄ ∈ V be the component of b̄ in the decomposition ann{a} ⊕ V of ann{a, ξ_{p}}. By Proposition 1, there exists γ̄ ∈ X_{1} ⊂ X^{G} such that

N_{E/F} γ̄ = {a} · v̄ = {a} · b̄ = β̄.

Then N_{E/F} (ᾱ−γ̄) = 0. By Lemma 4, ᾱ−γ̄ ∈ i_{E}(k_{n}F). But i_{E}(k_{n}F) ⊂ X^{G} + Y^{G} + Z. Hence ᾱ ∈ X^{G} + Y^{G} + Z and we have shown that (k_{n}E)^{G} = X^{G} + Y^{G} + Z.

For the dimension of Z, assume first that p > 2. By Theorem 5, N_{E/F} k_{n}E = ann_{k_{n}F} {a} and ker i_{E} = {a} · k_{n−1}F. Since {a, a} = 0 we see that ker i_{E} ⊂ N_{E/F} k_{n}E. Hence

d = dim_{𝔽_{p}} \frac{k_{n}F}{N_{E/F} k_{n}E} = dim_{𝔽_{p}} \frac{i_{E}(k_{n}F)}{i_{E}(N_{E/F} k_{n}E)} = dim_{𝔽_{p}} \frac{i_{E}(k_{n}F)}{Y^{G}},

where in the last equation we use Proposition 2 to identify Y^{G}. By Propositions 1 and 2, (X^{G} + Y^{G}) ∩ i_{E}(k_{n}F) = X_{2}^{G} ⊕ Y^{G}. Hence d = dim_{𝔽_{p}}(X_{2}^{G} ⊕ Y^{G} ⊕ Z)/Y^{G}. By Proposition 1, dim_{𝔽_{p}} X_{2}^{G} = Υ_{2}. Hence Υ_{2} + dim_{𝔽_{p}} Z = d for p > 2. Also we see that

dim_{𝔽_{p}} Z = dim_{𝔽_{p}} \frac{i_{E}(k_{n}F)}{X_{2}^{G} ⊕ Y^{G}} = dim_{𝔽_{p}} \frac{k_{n}F}{\{ξ_{p}\} · W + N_{E/F} k_{n}E} = z.

Now assume p = 2. By Propositions 1 and 2, i_{E}(k_{n}F)∩(X^{G}+Y^{G}) = Y^{G}. Proceeding as in the last case,

dim_{𝔽_{2}} Z = dim_{𝔽_{2}} \frac{i_{E}(k_{n}F)}{Y^{G}} = dim_{𝔽_{2}} \frac{i_{E}(k_{n}F)}{i_{E}(N_{E/F} k_{n}E)}

= dim_{𝔽_{2}} \frac{k_{n}F}{N_{E/F} k_{n}E + ker i_{E}}

= dim_{𝔽_{2}} \frac{k_{n}F}{N_{E/F} k_{n}E + \{a\} · k_{n−1}F} = z,

since ker i_{E} = {a} · k_{n−1}F, by Theorem 5.

We then consider the filtration

k_{n}F ⊃ (({a} · k_{n−1}F) + N_{E/F} k_{n}E) ⊃ N_{E/F} k_{n}E.

The dimension of the quotient of the first and third modules is, by definition, d. By Theorem 5, N_{E/F} k_{n}E = ann_{k_{n}F} {a}. Since {a, a} = {a, −1} we see that

({a} · k_{n−1}F) ∩ N_{E/F} k_{n}E = {a} · V.

Hence

dim_{𝔽_{2}} \frac{(\{a\} · k_{n−1}F) + N_{E/F} k_{n}E}{N_{E/F} k_{n}E} = dim_{𝔽_{2}} \frac{\{a\} · k_{n−1}F}{\{a\} · V} = dim_{𝔽_{2}}\{a\} · W.

By Lemma 2, dim_{𝔽_{2}}{a} · W = dim_{𝔽_{2}} W = Υ_{2}. Hence Υ_{2} + dim_{𝔽_{2}} Z = d for p = 2 as well. □

4. Proofs of Theorems 1 and 2

Proof of Theorem 1. By Propositions 1, 2, and 3, there exist independent submodules X = X_{1} + X_{2}, Y, and Z satisfying the conditions of the theorem. All that remains is to show that k_{n}E = X + Y + Z.

We proceed by induction on the length l(γ) of the cyclic submodule ⟨γ̄⟩ of k_{n}E generated by an arbitrary element γ̄ ∈ k_{n}E. If l(γ) = 1, then by Proposition 3, γ̄ ∈ X^{G} + Y^{G} + Z. Assume then that β̄ ∈ X + Y + Z if l(β) ≤ i < p and that l(γ) = i + 1.

Suppose first that l(γ) = 2 and

γ̄ ∈ {A} · i_{E}(k_{n−1}F) + (k_{n}E)^{G}.

Then (σ −1)γ̄ = i_{E}({ξ_{p}}·b̄) for some b ∈ K_{n−1}F. In the decomposition ann{a, ξ_{p}}⊕W of k_{n−1}F, write b̄ = f̄+w̄. By Proposition 1 there exists ω̄ ∈ X_{2} such that (σ−1)ω̄ = i_{E}({ξ_{p}}·w̄). We also have {ξ_{p}}·f̄ ∈ ann{a} and therefore by Theorem 5 and Proposition 2 there exists ȳ ∈ Y such that i_{E}(\{ξ_{p}\} · f̄) = i_{E}(N_{E/F} (ȳ)). Hence there exists \overline{y′} ∈ Y such that (σ − 1)γ̄ = (σ − 1)ω̄ + (σ − 1)\overline{y′}. Hence l(γ − ω − y′) ≤ 1 and by the inductive hypothesis γ̄ ∈ X + Y + Z.

Now since by the preceding arguments {A}·i_{E}(k_{n−1}F) ⊂ X +Y +Z, in order to show that an arbitrary γ̄ with l(γ) = 2 lies in X + Y + Z it is enough to show that γ̄ + {A} · i_{E}(b̄) ∈ X + Y + Z for any b ∈ K_{n−1}F.

Suppose then that l(γ) = 2 and

γ̄ ∉ {A} · i_{E}(k_{n−1}F) + (k_{n}E)^{G}.

Then, by Lemma 3, there exist b ∈ K_{n}F and α ∈ K_{n}E such that β̄ = γ̄ + {A} · i_{E}(b̄) satisfies l(β̄) ≤ 2 and ⟨β̄⟩^{G} = ⟨Nᾱ⟩. Hence (σ−1)^{l(β)−1}β̄ = cNᾱ for some c ∈ ℤ. But cNᾱ = i_{E}N_{E/F} (\overline{cα}) ∈ Y^{G}, by Proposition 2. Hence there exists ω̄ ∈ Y such that (σ − 1)^{p−1}ω̄ = cNᾱ. Now λ̄ = (σ − 1)^{p−l(β)}ω̄ ∈ Y and (σ − 1)^{l(β)−1}(β̄ − λ̄) = 0. Hence l(β − λ) < l(γ) and by the inductive hypothesis β̄ and hence γ̄ lie in X + Y + Z.

If l(γ) ≥ 3 then the same argument works again. By Lemma 3 ⟨γ̄⟩^{G} = ⟨Nᾱ⟩ and so (σ − 1)^{l(γ)−1}γ̄ = cNᾱ for some c ∈ ℤ. But cNᾱ = i_{E}N_{E/F} (cᾱ) ∈ Y^{G}, by Proposition 2. Hence there exists ω̄ ∈ Y such that (σ − 1)^{p−1}ω̄ = cNᾱ. Now λ̄ = (σ − 1)^{p−l(γ)}ω̄ ∈ Y and (σ − 1)^{l(γ)−1}(γ̄ − λ̄) = 0. Hence l(γ − λ) < l(γ) and by the inductive hypothesis γ̄ ∈ X + Y + Z. □

Proof of Theorem 2. By Propositions 1, 2, and 3, there exist independent submodules X = X_{1}, Y, and Z satisfying the conditions of the theorem. All that remains is to show that k_{n}E = X + Y + Z.

Let γ̄ ∈ k_{n}E be arbitrary. If l(γ) = 1, then by Proposition 3, γ̄ ∈ X^{G} + Y^{G} + Z. Otherwise (σ − 1)γ̄ = (σ + 1)γ̄ = i_{E}N_{E/F} γ̄ ∈ Y^{G}, by Proposition 2. Hence there exists ω̄ ∈ Y such that (σ − 1)ω̄ = (σ − 1)γ̄. Therefore l(γ − ω) < 2 and by the inductive hypothesis γ̄ ∈ X + Y + Z. □

5. Proof of Theorem 5

For the case p = 2 we have the long exact sequence of Galois cohomology groups due to Arason [A, Satz 4.5]. Suppose then that p > 2 and F is perfect. Let S be any p-Sylow subgroup of G_{F} = Gal(F_{sep}/F), and set L to be the fixed field of S. Because F is perfect, the separable closure F_{sep} is identical to the algebraic closure F̄, and hence each finite extension of L has degree a power of p. In particular, all of the hypotheses of Theorem 4 are valid for the field L in place of F. Furthermore, ([L : F], p) = 1. (Here we use basic properties of supernatural numbers and Sylow p-subgroups. See [Se, Chapter 1].) Therefore if E = F (\sqrt[p]{a}) is a cyclic extension of F of degree p, so is EL = L(\sqrt[p]{a}) over L. By Theorem 4 we see that the sequence

k_{m−1}EL ⟶^{N_{EL/L}} k_{m−1}L ⟶^{{a}·−} k_{m}L ⟶^{i_{EL}} k_{m}EL

is exact for each m ∈ ℕ.

We claim that i_{L}: k_{m}F → k_{m}L is injective. Indeed, suppose that i_{L}(α) = 0 for some α ∈ k_{m}F. Then there exists a finite subextension M/F of L/F such that i_{M}(α) = 0. Then

0 = N_{M/F} (i_{M}(α)) = [M : F]α,

(see [FV, p. 300]). Because [M : F] is coprime with p, we see that α = 0 and i_{L} is injective as asserted. Similarly we have that i_{EL}: k_{m}E → k_{m}EL is injective.

We then have the following commutative diagram:

\xymatrix{k_{m−1}EL \ar[r]^{N_{EL/L}} & k_{m−1}L \ar[r]^{\{a\}·−} & k_{m}L \ar[r]^{i_{LE}} & k_{m}EL \\ k_{m−1}E \ar[r]^{N_{E/L}} \ar[u]_{i_{EL}} & k_{m−1}F \ar[r]^{\{a\}·−} \ar[u]_{i_{L}} & k_{m}F \ar[r]^{i_{E}} \ar[u]_{i_{L}} & k_{m}E \ar[u]_{i_{EL}}}

Because the vertical maps are injective, we see that the bottom row of the diagram is a complex: the composition of any two consecutive maps is the zero map. We now establish exactness at the second and third terms of the complex.

Let α ∈ k_{m−1}F such that {a} · α = 0. Then {a} · i_{L}(α) = 0 and therefore there exists an element β ∈ k_{m−1}EL such that N_{EL/L}(β) = i_{L}(α). Let M/F be a finite extension such that β is defined over EM. Then N_{EM/M}(β) = i_{M}(α), and we have

N_{EM/F} (β) = N_{M/F} (N_{EM/M}(β)) = N_{M/F} (i_{M}(α)) = [M : F]α

and N_{EM/F} (β) = N_{E/F} (N_{EM/E}(β)). Thus

N_{E/F} (N_{EM/E}(β)) = [M : F]α.

Because ([M : F], p) = 1 we see that α ∈ N_{E/F} (k_{m−1}E). Therefore we have established the exactness of our complex at k_{m−1}F.

Now assume that α ∈ k_{m}F such that i_{E}(α) = 0 ∈ k_{m}E. Then arguing as above, we see that there exist a finite extension M/F and β ∈ k_{m−1}M such that

{a} · β = i_{M}(α) ∈ k_{m}M.

Applying N_{M/F} and using the projection formula we see that

{a} · N_{M/F} (β) = N_{M/F} (i_{M}(α)) = [M : F]α.

Because [M : F] is coprime with p, α ∈ {a} · N_{M/F} (γ) for a suitable element γ ∈ k_{m−1}M. Hence we see that our complex is also exact at k_{m}F and the full complex is exact. □

Remark. Assuming as usual that F contains a primitive root ξ_{p}, then if m ≤ 2, no further assumption on F in Theorem 5 is necessary. For m = 1 this claim follows from basic Kummer theory, and for m = 2 see [Me, Prop. 5] and [Sr, Chap. 5, Lemma 8.4].

6. Reduction to the Case char F = 0

Suppose now that char F = q > 0 and q ≠ p. We also assume that F is infinite, because if F is finite then K_{n}F = 0 for n ≥ 2 and therefore this is a trivial case. (See [FV, Prop. IX.1.3].) Assume as before that F contains a primitive pth root of unity ξ_{p} and E/F is a cyclic extension of degree p. We shall show that there exists an explicitly defined cyclic extension J/L of degree p such that char L = 0, so that L is perfect, and k_{n}J is naturally isomorphic with k_{n}E as a G = Gal(E/F) ≅ Gal(J/L) module.

Recall first that there exists a discrete valuation ring A of characteristic 0 such that its maximal ideal M is generated by q and A/M ≅ F. (Such a ring is called a q-ring. See [Ma, p. 223].) By passing to a completion Â of A with respect to M-adic topology and observing that Â/M̂ ≅ A/M ≅ F and M̂ = Â.q we see that we may and will assume that A is a complete local q-ring such that A/M ≅ F.

It is known that a complete q-ring is uniquely determined up to its isomorphism by its residue field. (See [Ma, Cor., p. 225].) Observe further that a complete discrete valued field is henselian. (See [R, Thm. 5].)

Now following [FW, §IX.3] we have a natural construction

R = \underrightarrow{lim}(A^{(1)} ⊂ A^{(2)} ⊂ A^{(3)} ⊂ ⋯), where
A^{(1)} = A as above and A^{(n+1)} := A^{(n)}[t]/(t^{p} − π_{n}),

where π_{n} is a uniformizer of A^{(n)} for n ≥ 1. As was noticed in [FW], this ring R is a henselian valuation ring of characteristic 0, with value group the underlying group of the ring ℤ[1/p].

For each i ∈ ℕ let L^{(i)} be the quotient field of A^{(i)}, and let L be the quotient field of R. Then from Lemma IX.3.5 in [FW], there is a natural isomorphism k_{n}F ≅ k_{n}L for each n ∈ ℕ.

Let T be the inertia subgroup of G_{L^{(1)}}. We have the natural isomorphism G_{L^{(1)}}/T ⟶^{∼} G_{F}. Let φ be the compositum of the natural surjections

G_{L^{(1)}} → G_{L^{(1)}}/T ⟶^{∼} G_{F} → Gal(E/F),

and let J^{(1)} be the fixed field of ker φ. Then Gal(J^{(1)}/L^{(1)}) is naturally isomorphic to Gal(E/F). (This is a special case of a more general construction about lifting certain Galois abelian extensions. See [K, Lemma 2.5].)

Thus we see that J^{(1)}/L^{(1)} is a cyclic, purely inert extension of degree p. Because the tower

L^{(1)} ⊂ L^{(2)} ⊂ ⋯ ⊂ L^{(n)} ⊂ ⋯

is a chain of totally ramified extensions L^{(n+1)}/L^{(n)}, we see that J^{(1)} ∩ L = L^{(1)}. Set J := J^{(1)}L. Then J/L is a cyclic, purely inert extension of degree p.

Now set J^{(i)} := J^{(1)}L^{(i)}, and let B^{(i)} be the unique valuation ring in J^{(i)} such that B^{(i)} ∩ L^{(i)} = A^{(i)}. (B^{(i)} is unique because A^{(i)} is henselian.) Then again following the proofs of Lemma IX.3.2 and Lemma IX.3.5 in [FW], we establish, in exactly the same way we proved that k_{n}F ≅ k_{n}L, that k_{n}E ≅ k_{n}J under a G-equivariant isomorphism. Our reduction is complete.

7. Acknowledgements

We are very grateful to M. Rost and to J.-P. Serre for correspondence concerning the Bloch-Kato conjecture. We are also very grateful to J.- P. Tignol for his kind and encouraging remarks concerning this paper.

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

E-mail address: nlemire@uwo.ca

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

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