Multiplicative Groups of Fields modulo Products of Subfields

J.-L. Colliot-Thélène, R. M. Guralnick^{1}, and R. Wiegand^{1}

1. Introduction

Let k be a field and K a proper finite separable field extension of k. Consider the quotient K^{∗}/k^{∗} of the multiplicative group K^{∗} of K by the multiplicative group k^{∗} of k. It is an easy exercise (see §4) to show that for any positive integer n, the n-torsion subgroup (set of elements killed by n) of K^{∗}/k^{∗} is a finite group. Also, if k is finitely generated over the prime field, classical results in arithmetical algebraic geometry imply that the whole torsion subgroup of K^{∗}/k^{∗} is a finite group.

Let now E_{i} (i = 1, …, r) be fields with k ⊂ E_{i} ⊂ K. Let K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} be the quotient of the multiplicative group K^{∗} of K by the subgroup spanned by the multiplicative groups E_{i}^{∗}. For r ≥ 2, are there finiteness results similar to those mentioned above for r = 1 ?

In this paper we show that this is indeed the case when r = 2 (Theorems 5.1 and 5.2, Corollary 5.3). Here are some of the results.

If E_{1}, E_{2} are subfields of K with k ⊂ E_{i} ⊂ K, then for any positive integer n, the n-torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} is finite, and it has no p-torsion for p = char(k).

If l is an odd prime, and k contains the l-th roots of unity, then the l-primary torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} is finite. The analogous result holds true for the 2-primary subgroup if the 4-th roots of unity are in k.

Let μ(K) denote the roots of unity in K. The torsion group of K^{∗}/E_{1}^{∗}E_{2}^{∗} is infinite if and only if μ(K)/μ(E_{1})μ(E_{2}) is infinite (Theorem 5.2). In particular, if k (or E_{i}) contains all roots of unity, then the torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} is finite.

If k is finitely generated over its prime subfield, then the whole torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} is finite.

For r ≥ 3, such finiteness statements no longer hold (see Example 6.4 or [GW, 1.2] for an example with r = 3 where the quotient group is an infinite group of exponent 2). For arbitrary r ≥ 1, for ℓ an odd prime number, when the ℓ-th roots of unity are in k, we show that the the ℓ-primary torsion subgroup of K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} is of bounded exponent (Corollary 6.2) and that this subgroup is finite if r = 2 (Corollary 5.3).

Similar (but weaker) finiteness results in the case r = 2 were first arrived at by recourse to Picard groups of singular curves ([W2] and [HW]).

The point of the present paper is to get at these finiteness results (and obtain the stronger ones mentioned above – in particular, results relating to the finiteness of the torsion subgroup and to the triviality of the p-torsion subgroup) in a direct fashion – essentially by means of Galois cohomology and such standard tools as Hilbert’s theorem 90 (of course, the results on fields finitely generated over the prime field also make use of deep classical results). Given k, K, E_{i} (i = 1, …, r) as above, L/k the Galois closure of K/k, and G = Gal(L/k), there are naturally associated G–modules of finite type S and T (see §2), and the quotient K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} is related to the Galois cohomology of some associated G-modules. For r = 2, the finiteness theorems follow from purely algebraic

1 partially supported by NSF facts: the module T is torsion-free and S is a permutation lattice (Theorem 2.6; see also Theorem 3.6).

The abelian groups K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} have also been studied in other papers ([B], [DM], [GW], [H], [S], [W1]). In those papers, the emphasis is on the torsion-free rank of the abelian group K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} (the starting point being Brandis’ result that if k is infinite, then the quotient K^{∗}/k^{∗} is not finitely generated). If r ≤ 2, then this quotient has infinite torsionfree rank (see [W1, 1.2], [GW, 5.5] or Theorem 5.10). If r ≥ 3, then there are examples with this quotient trivial, infinite torsion or of infinite torsionfree rank (see [GW] and §4, §6 and §7 below for various examples).

We first study the problem in the more general setting of a finite group acting on a module. Let G be a finite group and M a ZG-module. Let C be a family of subgroups C_{1}, …, C_{r} of G with D a subgroup of the intersection. Let M^{H} denote the fixed points of H on M. Let λ(M) = λ_{C}(M) = ∑_{i=1}^{r} M^{C_{i}} ⊆ M^{D}. We study the abelian group structure of M^{D}/λ(M) and in particular its torsion subgroup. If r = 2 and H = ⟨C_{1}, C_{2}⟩, we show that the structure of the torsion subgroup depends upon H^{1}(H, M) and the torsion subgroup of M. If both these groups are finite, then the torsion subgroup of M^{D}/λ(M) is finite (Corollary 2.8).

In §5 (for the case of two subfields) and §6 (for the general case), we use the group theoretic results (applied to Galois groups) together with cohomological information about fields to obtain the main results on fields mentioned above. In §7, we outline another approach to the case of fields finitely generated over the prime field.

In a sequel, we will explore the relationship between some of the results established here and Picard groups.

The second author would like to thank Al Weiss for some useful discussions.

2. Fixed Points on Modules

Fix a finite group G. Unless noted otherwise, all G-modules will be left G-modules and all tensor products will be over Z. Recall that if M and N are G-modules, then so are M ⊗ N (with g(m ⊗ n) = gm ⊗ gn) and Hom_{Z}(M, N) (with (gf)(m) = g(f(g^{−1}m)). If A is a subgroup of G, let X_{A} denote the ZG permutation lattice on the left cosets of A (note that X_{A} is just the induced module Z[G/A] = ZG ⊗_{ZA} Z. If g ∈ G, let ḡ denote the left coset gA. If M is a G-module, let M^{G} denote the fixed points of G on M. Note that if M is any G-module, then M^{A} and (M ⊗ X_{A})^{G} can be naturally identified as abelian groups via x → ∑ gx ⊗ ḡ, where the sum is over a set of coset representatives for the left cosets of A.

If A ⊆ B are subgroups of G, then there is a homomorphism φ = φ_{A,B} of G-sets from G/A onto G/B (given by φ(gA) = gB). This induces a homomorphism (which we also denote by φ) from X_{A} onto X_{B}. Let γ = γ_{A,B} : X_{B} → X_{A} be the map on the duals induced by φ (note any permutation lattice is self dual). Explicity, γ(gB) = ∑_{i=1}^{s} gb_{i}A, where b_{1}, …, b_{s} are left coset representatives for B/A. For example, if A = 1 and B = H, then the map Z[G/H] → Z[G] is the G-map which sends ¯1 to N_{H} := ∑_{h∈H} h. Note that γ is a split injection of abelian groups.

If M is any ZG-module, then there is an induced injection 1⊗γ : M ⊗X_{B} → M ⊗X_{A}. This induces a mapping on G-fixed points and via the identification above induces the inclusion M^{B} → M^{A}.

The maps γ and φ induce the short exact sequences

0 → X_{B} →^{γ} X_{A} → V → 0  (1)

and

0 → V′ → X_{A} →^{φ} X_{B} → 0.  (1′)

We note that V′ is a free abelian group of rank [G : A] − [G : B]. Since (1) and (1′) are dual to one another, the same is true for V.

Since (1) is split exact as a sequence of abelian groups, we can tensor with any ZGmodule M and obtain:

0 → M ⊗ X_{B} → M ⊗ X_{A} → M ⊗ V → 0.  (2)

Taking G-fixed points, and using the identification above and the isomorphism H^{1}(G, M ⊗ X_{B}) ≅ H^{1}(B, M) (Shapiro’s Lemma, cf [Br, Prop. 6.2. p. 73]), we get:

0 → M^{B} → M^{A} → (M ⊗ V)^{G} → H^{1}(B, M).  (3)

If we take (A, B) = (C, G) above, this yields:

Lemma 2.1. Let C be a subgroup of the finite group G. Let M be a G-module. Then M^{C}/M^{G} embeds into M^{s}, where s = [G : C] − 1.

Next we consider the situation of a family of subgroups. Let C = {C_{i} : 1 ≤ i ≤ r} be a family of subgroups of the finite group G. Let D ⊆ ∩_{i=1}^{r}C_{i} be a subgroup of G. Let X_{i} = X_{C_{i}} and γ_{i} = γ_{D,C_{i}}. Let γ : X_{1} ⊕ … ⊕ X_{r} → X_{D} be defined by γ(x_{1}, …, x_{r}) = γ_{1}(x_{1}) + … + γ_{r}(x_{r}). Set X = X_{1} ⊕ … ⊕ X_{r}.

This yields the basic exact sequence of G-modules

0 → S → X →^{γ} X_{D} → T → 0.  (4)

Here S = ker(γ) and T = coker(γ). Note that when r = 1, (4) is essentially (1). In particular, for r = 1, S = 0 and T is torsionfree.

Let Y = γ(X). It is convenient to break up sequence (4) into two short exact sequences:

0 → S → X →^{γ} Y → 0.  (5)
0 → Y → X_{D} → T → 0.  (6)

Since Y ⊆ X_{D}, the abelian group Y is free and so (5) is always split exact in the category of abelian groups.

Let M be a ZG-module. Let λ(M) = ∑_{i=1}^{r} M^{C_{i}} ⊆ M^{D}. We are interested in the structure of the abelian group M^{D}/λ(M). Let α = 1 ⊗ γ : M ⊗ X → M ⊗ X_{D}. Let N ⊆ M ⊗ X_{D} be the image of M ⊗ X under α. Since we are identifying (M ⊗ X_{i})^{G} with M^{C_{i}}, we see that α((M ⊗ X)^{G}) = λ(M) ⊆ N^{G} ⊆ (M ⊗ X_{D})^{G} = M^{D}. Thus

M^{D}/λ(M) ≅ coker(α : (M ⊗ X)^{G} → M^{D}).

Tensoring M with (5) yields a short exact sequence:

0 → M ⊗ S → M ⊗ X →^{α} M ⊗ Y → 0.  (7)

Taking fixed points yields:

0 → (M ⊗ S)^{G} → (M ⊗ X)^{G} →^{α} (M ⊗ Y)^{G} → H^{1}(G, M ⊗ S) → H^{1}(G, M ⊗ X) (8).

From now on, assume that one of the two abelian groups M and T is torsionfree, hence flat. If T is torsionfree, then (6) is split exact as a sequence of abelian groups and in fact N = M ⊗ Y —similarly, if M is torsionfree, then M ⊗ Y injects into M ⊗ X_{D} and so N = M ⊗ Y. Thus (8) yields:

0 → N^{G}/λ(M) → H^{1}(G, M ⊗ S) → ⊕_{i=1}^{r}H^{1}(C_{i}, M).  (8′)

Tensoring M with (6) yields the short exact sequence:

0 → N → M ⊗ X_{D} → M ⊗ T → 0.  (9)

Taking G-fixed points yields:

0 → N^{G} → M^{D} → (M ⊗ T)^{G} → H^{1}(G, N) → H^{1}(D, M),  (10)

and

0 → M^{D}/N^{G} → (M ⊗ T)^{G} → H^{1}(G, M ⊗ Y) → H^{1}(D, M).  (10′)

Set M_{1} = N^{G}. By (10′), M^{D}/M_{1} embeds into (M ⊗ T)^{G}. By (8′), M_{1}/λ(M) embeds into H^{1}(G, M ⊗ S). Thus:

Theorem 2.2. Let G be a finite group with subgroups C_{i}, 1 ≤ i ≤ r. Let D be a subgroup of ∩_{i}C_{i}. Let the G-modules S and T be defined as in (4). Let M be a ZG-module and assume that either M or T is Z-torsionfree. Let λ(M) = ∑_{i=1}^{r} M^{C_{i}} ⊆ M^{D}. Then there exists a subgroup M_{1} of M^{D} containing λ(M) such that

(a) M^{D}/M_{1} embeds into (M ⊗ T)^{G} ⊆ (M ⊗ T), and

(b) M_{1}/λ(M) embeds into H^{1}(G, M ⊗ S).

A G-module M is called coflasque if H^{1}(C, M) = 0 for all subgroups C of the finite group G. Permutation lattices are coflasque (by Shapiro’s Lemma). If M is any abelian group, let μ(M) = M_{tors} be its torsion subgroup.

Corollary 2.3. Keep notation and assumptions as in the previous theorem.

(a) If T is torsionfree and the torsion subgroup of M has exponent d, then the torsion subgroup of M^{D}/λ(M) has exponent dividing de, where H^{1}(G, M ⊗ S) has exponent e. Note that e always divides |G|.

(b) If T or M is torsionfree and H^{1}(G, M ⊗ S) = 0, then M^{D}/λ(M) embeds into (M ⊗ T)^{G} ⊆ (M ⊗ T). If in addition, T is torsionfree, then μ(M^{D}/λ(M)) embeds into (μ(M) ⊗ T)^{G}.

(c) If M and T are both torsionfree, then the torsion subgroup of M^{D}/λ(M) embeds into H^{1}(G, M ⊗ S).

(d) If M is torsionfree, then the torsion subgroup of M^{D}/λ(M) has exponent dividing te, where t is the exponent of the torsion subgroup of T and e is the exponent of H^{1}(G, M ⊗ S).

(e) If M is torsionfree and coflasque and T is torsionfree, then the torsion subgroup of M^{D}/λ(M) is isomorphic to H^{1}(G, M⊗S). In particular, if M = X_{C} for some subgroup C of G and T is torsionfree, then the torsion subgroup of M^{D}/λ(M) is isomorphic to H^{1}(C, S).

(f) If M = ZG, then M^{D}/λ(M) ≅ T.

Proof. (a) – (d) are immediate consequences of the previous theorem. Assume that both M and T are torsionfree and that M is coflasque. By (10′), M^{D}/N^{G} is torsionfree and so the torsion subgroup is isomorphic to N^{G}/λ(M). Since M is coflasque, H^{1}(G, M ⊗ X) = 0 and so (8′) yields N^{G}/λ(M) ≅ H^{1}(G, M ⊗ S). This proves the first assertion of (e). The last statement of (e) now follows by Shapiro’s Lemma.

Finally consider the case M = ZG. Since H^{1}(G, M ⊗ S) = 0, (8′) implies that N^{G} = λ(M). Since H^{1}(G, M ⊗Y) = 0, it follows from (10′) that M^{D}/λ(M) ≅ (M ⊗T)^{G} ≅ T.

In general, given a family of subgroups as above, T is not torsionfree. Moreover, there is little reason to expect that H^{1}(G, M ⊗S) should be 0. We note, however, that S satisfies:

Theorem 2.4. Let S be defined as in (4). Then H^{1}(G, S) = 0.

Proof. By taking fixed points on (5) and using H^{1}(G, X) = 0, we see that it suffices to show that γ(X^{G}) = Y^{G}. This in turn follows from the fact that γ_{i} induces a surjection of the rank-one submodule (X_{C_{i}})^{G} to (X_{D})^{G}.

We will show below (Example 2.9) that S need not always be coflasque (even if T is torsionfree). See Corollary 2.3(e) above and Theorem 7.2 for the relevance of this condition. When r = 1, S = 0. In the case of two subgroups, the groups S and T are still quite controlled, as we shall now see.

So keep notation as above and assume r = 2. Set H = ⟨C_{1}, C_{2}⟩. We will prove that in this case T is torsionfree and that S ≅ X_{H} (and in particular is coflasque). See §3 for a different proof. We first prove that T is torsionfree.

Proposition 2.5. Let Γ be a finite G-set. Set H = ⟨C_{1}, C_{2}⟩. Set s = t + u − v_{1} − v_{2}, where t = |Γ|, C_{i} has v_{i} orbits on Γ and H has u orbits on Γ.

(a) Let M be the ZG-permutation lattice on Γ. Then M/(M^{C_{1}} + M^{C_{2}}) is free of rank s.

(b) Let U be an abelian group. Let W be the abelian group of functions from Γ to U viewed as a G-module via its action on Γ. (So W = U × … × U and G acts via permuting the coordinates.) Then W/(W^{C_{1}} + W^{C_{2}}) ≅ U^{s}.

Proof. (a) We note that a subgroup N of a finitely generated free abelian group M is a direct summand if and only if d(p), the dimension of (N/N ∩ pM) over the prime field, is independent of the prime p. We apply this criterion to N = M^{C_{1}} + M^{C_{2}}.

Fix a prime p and set M̄ = M/pM. If L is a subgroup of M, let L̄ denote its image in M̄. Let π be the natural projection of M onto M̄.

We identify Γ with a permutation basis for M (i.e. it is a basis and is G-invariant). If C is a subgroup of G, then observe that a basis for M^{C} consists of ∑ ω, where the sum is over ω in a C-orbit of Γ. Observe that this is true for a permutation lattice over any commutative ring.

This implies that π(M)^{C} = π(M^{C}) and has dimension equal to f(C), where f(C) is the number of orbits of C on Γ (and in particular is independent of p).

Clearly M̄^{C_{1}} ∩M̄^{C_{2}} = M̄^{H} has dimension u. Thus π(N) = M̄^{C_{1}} +M̄^{C_{2}} has dimension v_{1} + v_{2} − u and in particular is independent of p. Hence N is a summand of M of rank v_{1} + v_{2} − u and M/N is free of rank t + u − v_{1} − v_{2}.

(b) Let M be the permutation lattice on Γ. Thus W = M ⊗ U where G acts trivially on U. If C is a subgroup of G, then W^{C} = M^{C} ⊗ U. Thus, since M^{C_{1}} + M^{C_{2}} is an abelian group summand of M, W^{C_{1}} + W^{C_{2}} = (M^{C_{1}} + M^{C_{2}}) ⊗ U. Hence W/(W^{C_{1}} + W^{C_{2}}) = [M/(M^{C_{1}} + M^{C_{2}})] ⊗ U ≅ U^{s} by (a).

The next result, for r = 2, identifies S and shows T is torsionfree.

Theorem 2.6. Let G be a finite group with subgroups D, C_{1} and C_{2} with D ⊆ C_{1} ∩ C_{2}. Let H = ⟨C_{1}, C_{2}⟩ ⊆ G. In the exact sequence

0 → S → X_{C_{1}} ⊕ X_{C_{2}} →^{γ_{1}+γ_{2}} X_{D} → T → 0

given by (4) for r = 2, T is torsionfree of rank s = s(C_{1}, C_{2}, D) := [G : D] + [G : H] − [G : C_{1}] − [G : C_{2}] and S ≅ X_{H} = Z[G/H].

Proof. Let S_{G} and T_{G} denote the corresponding terms in (4) when D = 1. Since X_{D} embeds into X_{E} for any E ≤ D via γ_{E,D} and since γ_{E,D} ∘ γ_{D,C_{i}} = γ_{E,C_{i}}, it follows that S = S_{G}. Similarly, we see that T_{D} embeds into T_{G}. So to show that T is torsionfree we may assume that D = 1 and so X_{D} = ZG.

The fact that T is torsionfree follows from Proposition 2.5(a) and Corollary 2.3(f) applied to M = ZG.

In this case, the image of γ_{i} is {∑_{g∈G} a_{g}g : a_{g} = a_{h} if gC_{i} = hC_{i}}. These are precisely the fixed points of C_{i} under the right regular representation of G on ZG. This shows that S ≅ γ_{1}(X_{1}) ∩ γ_{2}(X_{2}) is precisely the set of fixed points of H under the right representation of H on ZG. This is clearly isomorphic (under the left action) to the permutation lattice X_{H} as desired. In particular, S has rank [G : H] and so T = T_{D} has rank as given above.

Thus Theorem 2.2 always applies when r = 2. By Shapiro’s Lemma, H^{1}(G, M ⊗X_{H}) ≅ H^{1}(H, M). Theorem 2.2 gives :

Theorem 2.7. Let D, C_{1}, C_{2} be subgroups of the finite group G with D ⊆ C_{1} ∩ C_{2}. Set H = ⟨C_{1}, C_{2}⟩. There exists a short exact sequence of abelian groups

0 → U → M^{D}/(M^{C_{1}} + M^{C_{2}}) → V → 0,

where U is a subgroup of H^{1}(H, M) and V is a subgroup of M^{s}, where s is the integer defined in Theorem 2.6.

Corollary 2.8. Let D, C_{1}, C_{2} be subgroups of the finite group G with D ⊆ C_{1} ∩ C_{2}. Set H = ⟨C_{1}, C_{2}⟩.

(a) If H^{1}(H, M) = 0, then M^{D}/(M^{C_{1}} + M^{C_{2}}) embeds into M^{s}. If, moreover, M is torsionfree, then so is M^{D}/(M^{C_{1}} + M^{C_{2}}).

(b) If H^{1}(H, M) is finite and the torsion subgroup of M is finite, then the torsion subgroup of M^{D}/(M^{C_{1}} + M^{C_{2}}) is finite.

We will consider applications to fields in later sections. Let us just point out that Corollary 2.8(a) applies when M is the multiplicative group of a field and G is a finite group of field automorphisms.

We now give an example where S is not coflasque but T is torsionfree.

Example 2.9. Let G be an elementary abelian group of order 8 with generators x_{i}, 1 ≤ i ≤ 3. Set C_{i} = ⟨x_{i}⟩ and D = 1. Let C = ⟨y⟩, where y = x_{1}x_{2}x_{3}. Let S and T be defined by (4).

(a) T is an infinite cyclic group and g acts via multiplication by ρ(g), where ρ is the character on G with ρ(x_{i}) = −1 for all i.

(b) |H^{1}(C, S)| = 2.

Proof. Keep notation as in (4). Let Y be the image of γ. Consider the basis 1, x_{1}, x_{2}, x_{1}x_{2}, x_{3}, x_{1}x_{3}, x_{2}x_{3}, y for ZG, and the bases for X_{i} in compatible order. Then the matrix of γ with respect to these bases is

( \begin{matrix}1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \end{matrix} ) .

It is straightforward to compute that the rank of this matrix is 7 modulo p for every prime p (if p is odd, then Y/pY is complemented by the one dimensional irreducible representation with character ρ – if p = 2, then one computes directly that the matrix has rank 7). It follows that Y is an abelian group summand of ZG of rank 7. Thus T is infinite cyclic. Since the irreducible representation of G with character ρ is not a constituent of X, it must occur on T. Thus (a) holds.

Since T^{C} = 0, it follows that Y^{C} = (ZG)^{C}. A straightforward computation shows that |Y^{C}/γ(X^{C})| = 2 (indeed, to see that γ(X^{C}) ≠ Y^{C}, just observe that ϵ(γ(w)) ≡ 0

(mod 4) for every w ∈ X^{C}, but ϵ(1 + y) = 2, where ϵ : ZG → Z is the augmention map – thus, 1 + y ∈ Y^{C} − γ(X^{C})). Consider the sequence (5). Taking C fixed point yields:

0 → S^{C} → X^{C} →^{γ} Y^{C} → H^{1}(C, S) → H^{1}(C, X) = 0.

Thus, H^{1}(C, S) ≅ Y^{C}/γ(X^{C}) and (b) follows.

The next result shows that in some cases with r ≥ 3, S may be coflasque. A ZG-lattice is called invertible if it is a summand of a permutation lattice. Since permutation lattices are coflasque, so are invertible lattices.

Proposition 2.10. Let C_{1}, …, C_{r} be subgroups of G. Let S and T be defined by (4).

(a) Assume that G = C_{i}C_{j} for each i ≠ j. Then S is a trivial ZG-lattice of rank r − 1.

(b) If D = 1 and T = 0, then S is invertible.

Proof. (a) It suffices to prove that G acts trivially on S ⊗Q and that this vector space has dimension r − 1. The hypothesis implies that Q[G/C_{i}] and Q[G/C_{j}] have only the trivial irreducible constituent in common. Since each γ_{i} is injective, this implies that S ⊗ Q contains no nontrivial constituents. It remains only to compute the dimension. Since Q[G/D] has only a 1-dimensional fixed space which is in the image of γ, the result follows.

(4) splits for T = 0 and D = 1. Thus (b) follows.

Part (a) of the previous result applies when the C_{i} are any collection of maximal subgroups of a nilpotent group G or more generally any collection of pairwise nonconjugate maximal subgroups of a solvable group G (see [AG, Corollary 1]).

If the C_{i} are the collection of all nontrivial subgroups of G, then T is quite often zero. In particular, this is true for G any nonabelian simple group. See [GW, §4].

Lemma 2.11. Let M be a ZG-module. Let f : M → M ⊗ ZG be the injective G-module map defined by f(m) = ∑_{g∈G} m ⊗ g. Let F be any additive functor from G-modules to abelian groups. Then f induces the natural map π from F (M) to F (W) and ker(π) has exponent dividing |G|.

Proof. Let |G| = n. The composite G-map

Z → Z[G] → Z

(the first map is the norm, the second the augmentation map) is multiplication by n. Now tensor this sequence by any G-module M. The composite map is still multiplication by n. Since F is additive, the composite map F (M) → F (M ⊗ ZG) → F (M) is multiplication by n, and the kernel of the map F (M) → F (M ⊗ ZG) is killed by n.

Later (see Proposition 6.3) we will apply the previous result to the functor F defined by F (M) = M^{D}/λ(M).

We now investigate to what extent the torsion subgroup of M/λ(M) is determined by the torsion subgroup of M. We first prove an easy cohomological result. Recall that a group is locally P for some property P if every finitely generated subgroup has property P.

Lemma 2.12. Let G be a finite group. Let M be a ZG-module which is locally dgenerated as an abelian group. Then H^{1}(G, M) is finite and has order dividing |G|^{d}. If d = 1, then H^{1}(G, M) has finite order dividing the order of the largest solvable quotient of G.

Proof. First consider the case where G is cyclic of order m. Let g be a generator for G. Then it is straightforward to show that H^{1}(G, M) is isomorphic to a subgroup of the m-torsion subgroup of M/(g − 1)M. Since M is locally d-generated, the same is true of M/(g − 1)M. In particular, the m-torsion subgroup has order dividing m^{d}. Now consider the case that G is solvable. Let H be a normal subgroup of G. Recall the exact sequence (cf [Se, Prop. 4, p. 117])

0 → H^{1}(G/H, M^{H}) → H^{1}(G, M) → H^{1}(H, M)^{G}.

Let H be a maximal normal (proper) subgroup of G. If H is nontrivial, it follows by induction that R := H^{1}(G, M) has order divding m^{d}. If H is trivial, then G is cyclic and the result has just been proved.

Since R is a torsion group, it suffices to consider the p-primary subgroup R(p) of R. Then res : R(p) → H^{1}(P, M) is injective for a Sylow p-subgroup P of G (cf. [Br, 10.3, p. 84]) and the result follows by the result for p-groups.

Now assume d = 1 and let H be the maximal normal perfect subgroup of G (i.e. H is equal to its own commutator subgroup). Then G/H is the largest solvable quotient of G, and H acts trivially on M. So H^{1}(H, M) = Hom(H, M) = 0 and H^{1}(G, M) ≅ H^{1}(G/H, M). The result for d = 1 now follows from the general result.

Let G be a finite group with subgroups C_{1}, …, C_{r} and D ⊆ ∩_{i}C_{i}. Let S and T be defined as in (4). Let M be a G-module and let μ(M) denote its torsion subgroup. Set L = M/μ(M) and π : M → L. Let M_{i} = M^{C_{i}} and L_{i} = L^{C_{i}}. We conclude from the exact sequence

0 → μ(M_{i}) → M_{i} → L_{i} → H^{1}(C_{i}, μ(M))

that there exists a surjection V = ⊕_{i}V_{i} → (λ(L)/π(λ(M)) where V_{i} is a subgroup of H^{1}(C_{i}, μ(M)). In particular, (λ(L)/π(λ(M)) is a torsion group and so is a subgroup of J := (L^{D}/π(λ(M))_{tors}.

Set R equal to the cokernel of the natural map

f : μ(M^{D})/λ(μ(M)) → (M^{D}/λ(M))_{tors}.

So R ≅ (M^{D}/(λ(M)+μ(M^{D}))_{tors} is isomorphic (via the map induced by π) to a subgroup of J.

Now consider the short exact sequence

0 → λ(L)/π(λ(M)) → J → (L^{D}/λ(L))_{tors} → 0.  (∗)

The left-hand group is a homomorphic image of V (and so has exponent dividing |G|), and the right-hand group has exponent dividing t|G| where t is the exponent of the torsion subgroup of T (Corollary 2.3(d)). Thus J (and so R) has exponent dividing t|G|^{2}. So we have shown:

Proposition 2.13. Let G be a finite group with subgroups C_{i}, i = 1, …, r, and D ⊂ ∩_{i}C_{i}. Let M be a ZG-module with torsion subgroup μ(M). The natural map

f : μ(M^{D})/λ(μ(M)) → (M^{D}/λ(M))_{tors}

has cokernel of finite exponent dividing t|G|^{2} where t is the exponent of the torsion subgroup of T (as in (4)).

If T is torsionfree (e.g., if r ≤ 2), then we can obtain more precise formulations of the previous result. In (*) above, it follows by Corollary 2.3(c) that the right-hand group is a subgroup of H^{1}(G, L ⊗ S). This yields:

Theorem 2.14. Let G be a finite group with subgroups C_{i}, i = 1, …, r, and D ⊂ ∩_{i}C_{i}. Let H = ⟨C_{1}, …, C_{r}⟩. Let M be a ZG-module with torsion subgroup μ(M). Let S and T be defined by (4). Assume that T is torsionfree (e.g., r ≤ 2).

(a) The natural map

f : μ(M^{D})/λ(μ(M)) → (M^{D}/λ(M))_{tors}

has cokernel R of exponent dividing |G|^{2}, and there exists a short exact sequence

V → R → U → 0,

where V is a subgroup of ⊕_{i}H^{1}(C_{i}, μ(M)) and U is a subgroup of

H^{1}(G, (M/μ(M)) ⊗ S).

(b) If all the cohomology groups in (a) are finite, then |R| divides

|H^{1}(G, (M/μ(M) ⊗ S))||H^{1}(C_{1}, μ(M))| ⋯ |H^{1}(C_{r}, μ(M))|.

(c) If r = 2 and the cohomology groups in (a) are finite, then |R| divides

|H^{1}(C_{1}, μ(M))||H^{1}(C_{2}, μ(M))||H^{1}(H, M/μ(M)|.

(d) If r = 1, then R is a homomorphic image of H^{1}(C_{1}, μ(M)). In particular, if μ(M) is locally d-generated, R has order dividing |C_{1}|^{d}.

(e) If r = 1, D = 1 and C_{1} = G, then R ≅ ker(H^{1}(G, μ(M)) → H^{1}(G, M)).

Proof. (a) and (b) are immediate consequences of (*) and our previous observations. If r = 2, then T is torsionfree and S = X_{H}, whence (c) follows from (b). If r = 1, then (4) is essentially (1), and S = 0 and T is torsionfree. Thus (d) follows from (a), (b) and Lemma 2.12.

(e) Under the hypotheses of (e), R = J and L/λ(L) is torsionfree. Thus, by (*), R ≅ L^{G}/π(M^{G}) and the result follows from the long cohomology sequence.

We close this section with a simple remark about the torsionfree rank of M^{D}/λ(M). Since for this purpose we may replace M by Q ⊗ M, we may assume that M is a QGmodule. An immediate consequence of (8) and (10) is:

Proposition 2.15. Let M be a QG-module.

(a) M^{D}/λ(M) ≅ (M ⊗ T)^{G}.

(b) If T is infinite (e.g., if r ≤ 2) and M contains every irreducible QG-module, then M^{D}/λ(M) is nonzero.

(c) If T is infinite and M contains every reducible QG-module as a summand with infinite multiplicty, then M^{D}/λ(M) has infinite dimension over Q.

Proof. We first apply (10). Since M and N are QG-modules, (10) reduces to a short exact sequence. Thus M^{D}/N^{G} ≅ (M ⊗ T)^{G}. Since H^{1}(G, M ⊗ S) = 0, it follows from (8) that N^{G} = λ(M). Thus (a) holds.

Assume that T is infinite. Let V be an irreducible QG-summand of Q ⊗ T. By assumption, the dual V^{∗} of V is a QG-summand of M. Thus (V^{∗} ⊗ V)^{G} is a nonzero submodule of (M ⊗ T)^{G}. Now (b) follows from (a).

If r ≤ 2, then T is infinite (see Theorem 2.6—the part of the proof showing that T has positive rank is quite easy).

The proof of (c) is analogous to that of (b).

3. A Dual Approach

There is a dual version of some of the results of the previous section. Instead of considering γ : X → X_{D} in (4), we consider the dual map φ : X_{D} → X given explicitly by φ = (φ_{1}, …, φ_{r}), where φ_{i} = φ_{D,C_{i}}. This yields the exact sequence:

0 → T′ → X_{D} → X → S′ → 0.  (11)

Note that (11) is almost dual to (4). Indeed, they will be dual if S′ or T is torsionfree via the “duality” M → Hom_{Z}(M, Z). If T is torsionfree, then (4) is split exact as a sequence of abelian groups. Therefore applying this duality yields (11), which is also split exact as a sequence of abelian groups. Thus S′ is torsionfree. Similarly, if S′ is torsionfree, T is also torsionfree. So S′ is torsionfree if and only if T is torsionfree. In any case, the sequences become dual to one another after we tensor with Q.

We can recover some of the previous results by applying the functor Hom_{G}( . , M). Note that Hom_{G}(X_{i}, M) can be identified with M^{C_{i}}. Similarly Hom_{G}(X_{D}, M) = M^{D}. Thus, the map X_{D} → X induces a map from Hom_{G}(X, M) → Hom_{G}(X_{D}, M) and the cokernel of this map is M^{D}/λ(M).

Let Y′ = φ(X_{D}). It is convenient to break up (11) into short exact sequences:

0 → Y′ → X → S′ → 0,  (12)

and

0 → T′ → X_{D} → Y′ → 0.  (13)

Let M be a ZG-module. We apply the functor Hom_{G}( . , M) to (12) and (13). This yields:

0 → Hom_{G}(S′, M) → Hom_{G}(X, M) → Hom_{G}(Y′, M) → Ext_{G}^{1}(S′, M) → Ext_{G}^{1}(X, M)

(14)

and

0 → Hom_{G}(Y′, M) → Hom_{G}(X_{D}, M) → Hom_{G}(T′, M) → Ext_{G}^{1}(Y′, M) → Ext_{G}^{1}(X_{D}, M).

(15)

The identification mentioned above and (14) and (15) yield the analogue of Theorem 2.2. We can take M_{1} = Hom_{G}(Y′, M) in the next result.

Theorem 3.1. Let G be a finite group with subgroups C_{i}, 1 ≤ i ≤ r. Let D be a subgroup of ∩_{i}C_{i}. Let S′ and T′ be defined as in (11). Let M be a ZG-module. Then there exists a subgroup M_{1} of M^{D} containing λ(M) such that

(a) M^{D}/M_{1} embeds into Hom_{G}(T′, M),

(b) M_{1}/λ(M) embeds into Ext_{G}^{1}(S′, M), and

(c) the cokernel of the embedding in (b) is isomorphic to a subgroup of Ext_{G}^{1}(X, M).

If T (or equivalently S′) is torsionfree, then Theorem 3.1 and Theorem 2.2 give essentially identical information.

Corollary 3.2. If M is torsionfree, then the torsion subgroup of M/λ(M) is isomorphic to a subgroup of Ext_{G}^{1}(S′, M).

Proof. Let M_{1} be as in Theorem 3.1. If M is torsionfree, then M/M^{D} and Hom_{G}(T′, M) are both torsionfree. Thus, by 3.1(a) the torsion subgroup of M/λ(M) is contained in M_{1}/λ(M). The result follows from 3.1(b).

One case which appears easier to deal with in this guise occurs when T′ = 0 (or equivalently T is finite). We first record:

Lemma 3.3. The following conditions are equivalent:

(i) T′ = Kerφ = 0 in sequence (11); (ii) T in sequence (4) is finite; (iii) Im(γ) ∩ Z ≠ 0, where Z ⊂ X_{D} = Z[G/D] via the obvious map 1 ↦ ¯1.

Proof. The equivalence of (i) and (ii) follows from the fact that (4) and (11) become dual to one another after we tensor with Q. If T is finite, then Im(γ) has finite index in X_{D}. Thus (ii) implies (iii). Assume (iii) holds. Choose a nonzero integer n with n¯1 in the image of γ. Since ¯1 generates X_{D} as a G-module, this implies that the image of γ contains nX_{D} and so T is finite.

Corollary 3.4. Assume that γ : X → X_{D} has finite cokernel (or equivalently φ : X_{D} → X is injective).

(a) M^{D}/λ(M) is isomorphic to a subgroup of Ext_{G}^{1}(S′, M). In particular, it is a torsion group.

(b) If H^{1}(C_{i}, M) = 0 for each i, then M^{D}/λ(M) ≅ Ext_{G}^{1}(S′, M).

Proof. (a) follows immediately from Theorem 3.1 (since T′ = 0, M^{D} = M_{1} in the notation of 3.1). Since X_{i} is a torsionfree abelian group, Ext_{G}^{1}(X_{i}, M) ≅ H^{1}(G, Hom_{Z}(M, X_{i})) (cf [Br, p. 61]). Since Hom_{Z}(M, X_{i}) ≅ Hom_{Z}(M, Z)⊗X_{i} as G-modules, it follows by Shapiro’s Lemma that Ext_{G}^{1}(X_{i}, M) ≅ H^{1}(C_{i}, M) = 0. Thus Ext_{G}^{1}(X, M) = 0. It now follows from Theorem 3.1(c) that M^{D}/λ(M) ≅ Ext_{G}^{1}(S′, M).

Assume for the moment that D = 1 and the family {C_{i}} is the collection of all nontrivial subgroups of G (or all subgroups of prime order if G is not of prime order). This case has been studied in [GW] and [H]. Corollary 3.4(b) for this case is essentially contained in [H, Prop. 3.5].

Recall that a finite group G is called a Frobenius complement if there exists a nonzero QG-module V such that no nontrivial element of G fixes a nonzero element of V. In [GW, Theorem 2.2] it was shown that T is torsion (or equivalently T′ is zero) if and only if G is not a Frobenius complement. In the notation of [GW, §1], I(G) is the image of γ. Note that if the positive integer s ∈ Z ⊂ Z[G] is in I(G), then s = ∑ a_{C}N_{C}, where the sum is over all subgroups of prime order in G, a_{C} ∈ ZG and N_{C} = ∑_{c∈C} c. Applying this to any element m in the G-module M yields sm ∈ λ(M) for all m ∈ M. Moreover, [GW, Theorem 3.1], when G is not a Frobenius complement, either T = 0 (i.e. I(G) = ZG and M = λ(M) for every M) or there exists a prime p such that T has exponent p or p^{2} (and so the same is true for M/λ(M)). Note that if T = 0, then (4) splits as a sequence of G-modules (since D = 1). Thus S is a summand of a permutation lattice and is coflasque. In fact, one can show by a minor variation of the proof of Theorem 2.4 that for this collection of subgroups, S is always coflasque.

As in section 2, the case of at most two subgroups is particularly nice. If there is only one subgroup, one can pass freely between the two approaches by taking duals. See (1) and (1′). We first record:

Lemma 3.5. Let A be a ring (with 1). Let

P →^{f} Q →^{g} R

be a complex of A-modules. If, for every A-module M, the induced complex of abelian groups

Hom_{A}(R, M) → Hom_{A}(Q, M) → Hom_{A}(P, M)

is exact, then the original complex is an exact sequence.

Proof. Let M be any injective module containing N = ker(g)/f(P) (see [CE, p. 9]). Assume that N ≠ 0 (and so M ≠ 0). Since M is injective there exists h : Q → M with ker(h) ∩ ker(g) = f(P). Thus h ∘ f = 0, but h ≠ w ∘ g for any w ∈ Hom_{A}(R, M) (since h does not vanish on all of ker(g)). Thus the induced complex above with this M is not exact.

We now show that for the case of two subgroups (11) has a very nice form. In particular, S′ is torsionfree and the sequence is split exact as a sequence of abelian groups. Dualizing then yields a different (and perhaps easier) proof of Theorem 2.6. It is curious to note, however, that the dual approach does not seem to yield Theorem 2.4.

Theorem 3.6. Let G be a finite group with subgroups C_{1}, C_{2} and D ⊆ C_{1} ∩ C_{2}. Let H = ⟨C_{1}, C_{2}⟩. The exact sequence (11) is now:

0 → T′ → Z[G/D] → Z[G/C_{1}] ⊕ Z[G/C_{2}] → Z[G/H] → 0.  (16)

In particular, S′ is torsionfree.

Proof. We have the obvious complex of G-modules

Z[G/D] → Z[G/C_{1}] ⊕ Z[G/C_{2}] → Z[G/H],

where the first maps send the class gD to (gC_{1}, gC_{2}) and the second map sends (aC_{1}, bC_{2}) to aH − bH. Note that the last map is obviously surjective.

For any G-module M, the induced sequence via Hom_{G}( . , M) is none other than

M^{H} → M^{C_{1}} ⊕ M^{C_{2}} → M^{D}

with a suitable minus sign somewhere, and that sequence is obviously exact. It remains only to apply Lemma 3.5 to conclude that the sequence above is exact. All in all, this gives us the exact sequence (16) above (with the modification of a minus sign).

We can also prove part of Proposition 2.5(a) from this approach. Simply start with the result of Theorem 2.6, and use H^{1}(G, Z[G/H] ⊗ M) = 0 if M is a permutation lattice. One then gets M^{D}/(M^{C_{1}} + M^{C_{2}}) ⊆ Hom_{G}(T′, M), which shows the left-hand quotient is free.

4. The One Subfield Case

If A is a ring with 1, let A^{∗} be its group of units. If K is any field, let μ(K) be the group of roots of unity in K. Let μ_{n} denote the group of n^{th} roots of unity in an algebraic closure of K. Let μ_{n}(K) be the group of n^{th} roots of unity in K.

We consider the structure of the torsion group of K^{∗}/F^{∗} for K a finite proper separable field extension of F. Let L be the Galois closure of K/F. Lemma 2.1, applied to the module L^{∗} with G the Galois group of L/F, yields the following:

Proposition 4.1. Let K/F be a finite separable extension of fields. Let L be the Galois closure of K/F.

(a) K^{∗}/F^{∗} embeds into a direct product of [K : F] − 1 copies of L^{∗}.

(b) The n-torsion subgroup is finite for all n, and there is no p-torsion if K has positive characteristic p.

(c) If μ(L) is finite, then the torsion subgroup of K^{∗}/F^{∗} is finite.

(d) If F is finitely generated over the prime field, then the torsion subgroup of K^{∗}/F^{∗} is finite.

If L is finitely generated over its prime field P, then so is any subfield. In particular, the subfield P (μ(L)) is finitely generated and algebraic over P. This implies that μ(L) is finite (if the characteristic is positive, this subfield is finite; if the characteristic is zero, this follows from the fact the degrees of the cyclotomic polynomials tend to infinity). This yields the last assertion in Proposition 4.1.

It may happen that the torsion subgroup of K^{∗}/F^{∗} is infinite. By Proposition 4.1 this can happen only when μ(L) is infinite. In particular, if K = C and F = R, then the torsion subgroup of K^{∗}/F^{∗} infinite. We now construct another example. Let ℓ be a prime, and let K = Q(ℓ^{∞}) be the subfield of C generated by ℓ^{d}-th roots of 1 for all d. The automorphism group of K is the multiplicative group of units in the ℓ-adic integers (this is because the cyclotomic polynomials are irreducible over Q). In particular, there exists an automorphism of order ℓ − 1. If ℓ is odd, let F be the subfield fixed by this automorphism. Then F does not contain a root of unity of order ℓ. We conclude that the (infinite) ℓ-primary torsion subgroup of K^{∗} injects into the ℓ-primary torsion subgroup of K^{∗}/F^{∗}. If ℓ = 2, let F be the fixed field of complex conjugation. Then −1 is the only root of unity in F and the map from the 2-primary torsion subgroup of K^{∗} to the 2-primary torsion subgroup of K^{∗}/F^{∗} has kernel of order 2. Again, K^{∗}/F^{∗} has an infinite 2-primary torsion subgroup. There are analogous examples with fields of positive characteristic.

We note that an example like the one above is impossible if l is odd and μ_{ℓ} ⊆ F (μ_{4} ⊆ F when ℓ = 2):

Lemma 4.2. Let L/F be a finite extension of fields of degree n. If ℓ is an odd prime and μ_{ℓ} ⊂ F, or if ℓ = 2 and μ_{4} ⊂ F, then the ℓ-primary subgroup of μ(L)/μ(F) is finite and has order dividing n.

Proof. Let U be the ℓ-primary subgroup of μ(F) and V the ℓ-primary subgroup of μ(L). If U is infinite, then U = V and the result holds. So assume U is finite. Let η ∈ V − U. It suffices to show that η^{n} ∈ U. Let m be the smallest power of ℓ such that η^{m} ∈ U. By hypothesis, η^{m} ≠ ±1. It follows by [K, Theorem 51] that x^{m} − η^{m} is irreducible over F. Thus m|n.

Let G be a finite group of automorphisms of the field L. We need to know about the cohomology of the module L^{∗}/μ(L). Since H^{1}(G, L^{∗}) = 0 by Hilbert’s theorem 90, the long exact sequence for cohomology yields an injection:

0 → H^{1}(G, L^{∗}/μ(L)) → H^{2}(G, μ(L)).

Lemma 4.3. Let L be a field of characteristic p ≥ 0. Assume G is a finite group of automorphisms of L.

(a) H^{1}(G, L^{∗}/μ(L)) is finite and its order is not divisible by p.

(b) For i > 0, H^{i}(G, μ(L)) is finite and has order not divisible by p.

(c) H^{1}(G, μ(L)) has order dividing the order of the largest solvable quotient of G.

Proof. By the sequence preceding the lemma, (b) implies (a). We now prove (b). Let ℓ be a prime and let U_{ℓ} be the ℓ-primary component of μ(L). Let Z_{ℓ} be the ring of ℓ-adic integers and Q_{ℓ} its quotient field. If ℓ does not divide the order of G, then H^{i}(G, U_{ℓ}) is a Z_{ℓ}-module annihilated by |G| and so is 0. If p > 0, then U_{p} = 0 and H^{i}(G, U_{p}) = 0. Since μ(L) is the direct sum of its primary subgroups, it suffices to prove that H^{i}(G, U_{ℓ}) is finite for each prime ℓ ≠ p dividing the order of G. For each prime ℓ, the ℓ-primary component of μ(L) is either a finite (cyclic) group or is isomorphic (as abelian groups) to the divisible group Q_{ℓ}/Z_{ℓ}. Since H^{i}(G, M) is finite whenever M is a finitely generated G-module, we may assume that U_{ℓ} ≅ Q_{ℓ}/Z_{ℓ}.

Consider the exact sequence of Z_{ℓ}G-modules

1 → Z_{ℓ}(1) → Q_{ℓ}(1) → Q_{ℓ}(1)/Z_{ℓ}(1) → 1,

where Z_{ℓ}(1) is the projective limit of the Z_{ℓ}-modules μ_{ℓ^{n}}(L) (with respect to the ℓ^{th} power map), Q_{ℓ}(1) = Q ⊗ Z_{ℓ}(1) and Q_{ℓ}(1)/Z_{ℓ}(1) is the quotient, isomorphic to the group of ℓ-primary roots of 1. (All ℓ-primary roots of 1 are in L since we are assuming U_{ℓ} is infinite.) As a sequence of Z_{ℓ}-modules, this sequence is isomorphic to

0 → Z_{ℓ} → Q_{ℓ} → Q_{ℓ}/Z_{ℓ} → 0.

But for i ≥ 1, H^{i}(G, Q_{ℓ}(1)) = 0 since this group is killed by the order of G and is also a Q_{ℓ}-vector space. Hence by the cohomology exact sequence

H^{i}(G, Q_{ℓ}(1)/Z_{ℓ}(1)) ≅ H^{i+1}(G, Z_{ℓ}(1))

for i ≥ 1, and the right-hand group is a finitely generated Z_{ℓ}-module killed by the order of G, hence finite.

Since μ(L) is locally cyclic, (c) follows by Lemma 2.12.

The next result shows that infinite torsion in K^{∗}/F^{∗} occurs precisely because of roots of unity.

Theorem 4.4. Let K/F be a finite proper separable extension of fields with Galois closure L. Let G be the Galois group of L/F.

(a) The natural injection f : μ(L)/μ(F) → (L^{∗}/F^{∗})_{tors} has cokernel isomorphic to the finite group H^{1}(G, μ(L)).

(b) The natural injection h : μ(K)/μ(F) → (K^{∗}/F^{∗})_{tors} has finite cokernel isomorphic to a subgroup of the finite group H^{1}(G, μ(L)).

(c) The torsion subgroup of K^{∗}/F^{∗} is finite if and only if μ(K)/μ(F) is finite.

(d) If μ(K) is finite, then the torsion subgroup of K^{∗}/F^{∗} is finite. In particular, this is true if F is finitely generated over the prime field.

Proof. (a) follows from Theorem 2.14(e) and Hilbert’s Theorem 90. (b) follows from Theorem 2.14(d). Now (b) implies (c) and (d).

We now have:

Corollary 4.5. Let K/F be a finite separable extension of fields. Let L be the Galois closure of K/F. Let F_{1} = F [μ(K)] and F_{2} be the maximal solvable extension of F contained in L.

(a) If ℓ is an odd prime and μ_{ℓ} ⊂ F, then the ℓ-primary subgroup of K^{∗}/F^{∗} is finite and has order dividing [F_{1} : F][F_{2} : F].

(b) If μ_{4} ⊂ F, then the 2-primary subgroup of K^{∗}/F^{∗} is finite and has order dividing [F_{1} : F][F_{2} : F].

Proof. Let ℓ be a prime and set U to be the ℓ-primary torsion subgroup of K^{∗}. Note that U ⊂ F_{1}. If (a) holds with ℓ odd or (b) with ℓ = 2, it follows by Lemma 4.2 that U/(U ∩F^{∗}) has order dividing [F_{1} : F]. Theorem 4.4 asserts that the cokernel of the map U/(U ∩ F^{∗}) into the ℓ-primary torsion subgroup of K^{∗}/F^{∗} has order dividing |H^{1}(G, μ(L))|. By Lemma 4.3(c), this latter group has order dividing [F_{2} : F]. This proves (a) and (b).

Let K/F be as above. Brandis [B] proved that if K is infinite, then K^{∗}/F^{∗} is not finitely generated. Davis and Maroscia [DM] proved that it has infinite torsionfree rank unless K is algebraic over a finite field or K/F is purely inseparable. One reduces very quickly to the case that K/F is a finite separable extension. See also [RV] for results about embeddings of tori. As we mentioned in the introduction (see also 6.7 below), under this hypothesis the analogous result is true for two subfields as well, but not necessarily true for three. If E_{i}, 1 ≤ i ≤ r, are proper intermediate subfields of the finite separable extension K/F with K not algebraic over a finite field, then K^{∗}/E_{1}^{∗}E_{2}^{∗} has infinite torsionfree rank; but there are examples where K^{∗}/E_{1}^{∗}E_{2}^{∗}E_{3}^{∗} is trivial [GW, 1.11], finite but nontrivial (see Example 6.6 below), infinite torsion [GW, Theorem 1.5] or of infinite torsionfree rank.

5. Two Subfields

By Hilbert’s Theorem 90, we know that if G is a finite group of automorphisms of the field L, then H^{1}(G, L^{∗}) = 0. Thus Theorem 2.7 and Corollary 2.8 yield:

Theorem 5.1. Let K/k be a finite separable extension of fields. Let E_{1} and E_{2} be intermediate fields with F = E_{1} ∩ E_{2}. Let L be the Galois closure of K/k.

(a) For each positive integer n, the n-torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} is finite and has order dividing |μ_{n}(L)|^{s}, where s = [K : F] + 1 − [E_{1} : F] − [E_{2} : F].

(b) If μ(L) is finite, then the torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} is finite.

(c) There is no p-torsion in K^{∗}/E_{1}^{∗}E_{2}^{∗} if k has positive characteristic p.

(d) If k is finitely generated over the prime field, then the torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} is finite.

Proof. Let G be the Galois group of L/F. Let C_{i} = Gal(L/E_{i}) and D = Gal(L/K). Since E_{1} ∩ E_{2} = F, G = ⟨C_{1}, C_{2}⟩. We apply Corollary 2.8 (note G = H in the notation of Corollary 2.8) to the G-module L^{∗}. Since H^{1}(G, L^{∗}) = 0, it follows that K^{∗}/E_{1}^{∗}E_{2}^{∗} embeds into (L^{∗})^{s}. Since μ_{n}(L) is cyclic and has no elements of order p if p is the characteristic of k, (a), (b) and (c) follow trivially. As for (d), it follows from (b) (since L is finitely generated over the prime field, μ(L) is finite).

Wiegand [W, Corollary 2.4] proved the finiteness result for (a) when n is not a multiple of the characteristic. Holley [HW] was the first to prove Theorem 5.1(a), (b) and (d) for n not divisible by the characteristic of k. Both Wiegand and Holley depend upon the finiteness of the n-torsion in Picard groups of curves [W, Theorem 2.1]. Their results give no information on the finiteness of the p-torsion when the field has positive characteristic p.

We can actually do a bit better than Theorem 5.1 in estimating the size of the torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗}.

Theorem 5.2. Let K/k be a finite separable extension of fields. Let E_{1} and E_{2} be intermediate fields. Let L be the Galois closure of K/k and G the Galois group of L/k. Then the natural map

α : μ(K)/μ(E_{1})μ(E_{2}) → (K^{∗}/E_{1}^{∗}E_{2}^{∗})_{tors}

has finite kernel and cokernel. The cokernel has exponent dividing |G|^{2}. The kernel is cyclic of order dividing |H^{1}(G, μ(L))|. In particular, its order divides |G|.

Proof. The fact that the cokernel has exponent dividing |G|^{2} follows from Proposition 2.13 (see also 2.14 (a)) and from the fact that T is torsionfree. The finiteness of the cokernel follows from Lemma 4.3 and Theorem 2.14(c).

Now we prove that the kernel is finite. Let F = E_{1} ∩ E_{2}. If x ∈ μ(K) corresponds to an element of the kernel of α, then x = yz with y ∈ E_{1} and z ∈ E_{2}. Suppose x^{n} = 1. Thus y^{n} = z^{−n} ∈ F and so y is a torsion element in E_{1}^{∗}/F^{∗} and z is a torsion element in E_{2}^{∗}/F^{∗}. By Theorem 4.4, the torsion subgroup of (E_{i}^{∗}/F^{∗}μ(E_{i})) has finite order d_{i} dividing |H^{1}(G, μ(L))|. Let d be the least common multiple of the d_{i}. Then y^{d} ∈ μ(E_{1})F^{∗} and z^{d} ∈ μ(E_{2})F^{∗}, and so x^{d} = y^{d}z^{d} ∈ (μ(E_{1})μ(E_{2})F^{∗}) ∩ μ(K) = μ(E_{1})μ(E_{2}). Thus the kernel has finite exponent dividing |H^{1}(G, μ(L))|, which divides |G| by Lemma 4.3(c).

The kernel is a subgroup of a homomorphic image of μ(K). Every finitely generated subgroup of μ(K) is cyclic and so the same is true for the kernel. Thus every finitely generated subgroup of the kernel is cyclic of order dividing d, and so the same is true for the entire kernel.

Corollary 5.3. Let K/k be a finite separable field extension. Let E_{1} and E_{2} be subfields of K each containing k.

(a) If μ(K)/μ(E_{1})μ(E_{2}) is finite, then the torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} has finite order not divisible by the characteristic of k. In particular, this is true if k is finitely generated over the prime field.

(b) If k contains μ_{ℓ} (ℓ odd) or μ_{4} (ℓ = 2), then the ℓ-primary torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} is finite.

Proof. The finiteness statement in (a) is an immediate consequence of the previous theorem. The fact that the order is not divisible by the characteristic of k follows from Theorem 5.1. (b) follows from Theorem 5.2 and Lemma 4.2.

A version of Theorem 5.1 holds in a more general situation. While Galois theory does not apply directly to (commutative) semisimple rings, it does to an extent that is sufficient for our purposes. Since the approach is so elementary, we include the next result. Most of it could be derived from the more general Galois theory of semiprime or artinian rings (cf [Kh] and [N]). If K is a commutative finite-dimensional algebra over the field k, then K/k is separable if and only if K is a direct product of fields each of which is separable over k.

Proposition 5.4. Let k be a field. Let K be a finite-dimensional separable k-algebra.

(a) There exists a finite-dimensional extension L of k containing K such that L^{G} = k, where G = Aut_{k}(L) with G finite.

(b) If E is a k-subalgebra of L, then there exists a subgroup C of G such that L^{C} = E.

(c) If H is a subgroup of G, then H^{1}(H, L^{∗}) is finite and its order is not divisible by the characteristic of k. Indeed, if we assume that H acts transitively on the primitive idempotents of L, then H^{1}(H, L^{∗}) is isomorphic to a subgroup of the abelianization of a subgroup of H.

(d) Let μ(L) denote the torsion subgroup of L^{∗}. Then H^{1}(H, L^{∗}/μ(L)) is finite and its order is not divisible by the characteristic of k.

Proof. Let e_{i}, 1 ≤ i ≤ d be the primitive idempotents of K. Let K_{i} = Ke_{i}. We may assume that each K_{i} is contained in a given algebraic closure of k. Choose F a finite dimensional Galois extension of k containing each K_{i}. Set L = F e_{1} × … × F e_{d}. Then G = Aut_{k}(L) = Aut_{k}(F) ≀ S_{d} is finite and L^{G} = k. Thus (a) holds.

We now prove (b) by induction on d. If d = 1, the result follows by the Galois correspondence theorem. If E contains nontrivial idempotents, then L decomposes with respect to these idempotents and the result follows by induction. So we may assume that E is a field.

Since E is a field, it is isomorphic (over k) to its first projection E_{1}. Thus, there exist σ_{i} ∈ Aut_{k}(F) (with σ_{1} the identity) such that

E = {(σ_{1}(u), …, σ_{d}(u)) : u ∈ E_{1}}.

Let C be the subgroup of G fixing E pointwise. If ρ ∈ G is any element acting via permutation of the coordinates and τ_{i} ∈ Aut_{k}(F), then ρ(τ_{1}, …, τ_{d}) fixes E pointwise if and only if τ_{ρ(i)}σ_{ρ(i)} = σ_{i} on E_{1}.

Let E′ be the subalgebra fixed pointwise by C. By the previous paragraph, C acts transitively on the set of primitive idempotents of L and so E′ must be a field. Thus E′ ⊆ F′ := {(σ_{1}(u), …, σ_{d}(u)) : u ∈ F }, which is k-isomrophic to F.

By the remarks above, the group J of automorphisms which stabilize F′ induces the full group of automorphisms Aut_{k}(F′) on F′ and by the usual Galois correspondence, C ∩J is trivial on no larger subfield of F′ than E. Thus E = E′ as desired.

It remains only to prove the cohomology statements. Clearly, we may assume that H is transitive on the primitive idempotents of L. Let B be the subgroup of H stabilizing a primitive idempotent e of L. Then Le ≅ F and L^{∗} is isomorphic to the induced module (F^{∗})_{B}^{H}. Hence, by Shapiro’s Lemma, H^{1}(H, L^{∗}) ≅ H^{1}(B, F^{∗}). Let A be the subgroup of B acting trivially on F. Since H^{1}(B/A, F^{∗}) = 0 (by Hilbert’s Theorem 90), it follows that H^{1}(B, F^{∗}) embeds into Hom_{B}(A, F^{∗}) = Hom_{B}(A, μ(F)). Since μ(F) is a locally cyclic group containing no elements of order equal to the characteristic of k, (c) follows.

Let M = L^{∗}/μ(L). As above, H^{1}(H, M) = H^{1}(B, F^{∗}/μ(F)). Since F^{∗}/μ(F) is torsionfree, it follows that H^{1}(B, F^{∗}/μ(F)) ≅ H^{1}(B/A, F^{∗}/μ(F)). This latter group is a finite group whose order is not divisible by the characteristic of k by Lemma 4.3.

If K/k is a finite-dimensional separable extension, a Galois closure of K/k is a separable finite-dimensional extension L/k such that K ⊆ L, L^{G} = k for G = Aut_{k}(L), and no proper k-subalgebra of L satisfies those two conditions. Note that the previous result shows that Galois closures exist (and there is a uniqueness statement as well which we do not need). We can now apply the methods used to consider the field case. If K/k is a finite dimensional separable extension and E is an intermediate k-algebra, we can view E^{∗} as the fixed points of a finite group G acting on L^{∗}, where L is a Galois closure of K/k. This module has most of the properties as in the field case (in particular, although H^{1}(G, L^{∗}) need not be zero, it is still finite, the m-torsion subgroup of L^{∗} is finite, every finitely generated subgroup of μ(L) is generated by at most d elements (where d is the number of primitive idempotents in K), and H^{1}(G, L^{∗}/μ(L)) is finite). These results in conjunction with the proofs above yield:

Theorem 5.5. Let k be a field and K/k a finite-dimensional separable extension. Let E_{1} and E_{2} be k-subalgebras of K. Let J be the torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} and J_{n} the elements of J whose order divides n. Then

(a) J_{n} is finite.

(b) J_{p} = 0, if k has positive characteristic p.

(c) If μ(K) is finite, then J is finite.

(d) If k is finitely generated over the prime field, then J is finite.

(e) The natural map μ(K)/μ(E_{1})μ(E_{2}) → (K^{∗}/E_{1}^{∗}E_{2}^{∗})_{tors} has finite cokernel.

The example with K = C × C and k the diagonal shows that even over algebraically closed fields we cannot expect finite torsion in K^{∗}/k^{∗}.

If we replace the assumption of separability by semisimplicity (i.e., K is a finitedimensional commutative k-algebra which is a direct product of fields), most of the previous results apply. Assume k has positive characteristic p. If F is k-subalgebra of K, let F_{s} denote the subalgebra of k-separable elements of F. Note F^{∗}/F_{s}^{∗} is a p-group of finite exponent (because of the finite dimensionality). Then, K_{s}^{∗}/(E_{1})_{s}^{∗}(E_{2})_{s}^{∗} maps into K^{∗}/E_{1}^{∗}E_{2}^{∗}. Since the group on the left has no elements of order p, it follows that this map is an embedding. The cokernel is a p-group of finite exponent. Thus K^{∗}/E_{1}^{∗}E_{2}^{∗} is the direct sum of a p-group of finite exponent and K_{s}^{∗}/(E_{1})_{s}^{∗}(E_{2})_{s}^{∗}. This latter group satisfies the conclusions of Theorem 5.5. Thus:

Corollary 5.6. Let k be a field of characteristic p > 0 and K/k a finite-dimensional semisimple commutative extension. Let E_{1} and E_{2} be k-subalgebras of K. Let J be the torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} and J_{n} the elements of J whose order divides n. Let J_{p′} denote the subgroup of J consisting of elements whose order is not divisible by p. Then (a) J_{n} is finite for n prime to p.

(b) The p-primary subgroup of J has finite exponent.

(c) If μ(K) is finite, then J_{p′} is finite.

(d) If k is finitely generated over the prime field, then J_{p′} is finite.

(e) The natural map μ(K)/μ(E_{1})μ(E_{2}) → J_{p′} has finite cokernel.

6. Several Subfields

If there are more than two subfields, we cannot say nearly as much. Applying Proposition 2.13 yields:

Theorem 6.1. Let E_{i}, 1 ≤ i ≤ r, be intermediate fields for the finite separable field extension K/k. The cokernel of the natural map

μ(K)/μ(E_{1}) … μ(E_{r}) → (K^{∗}/E_{1}^{∗} … E_{r}^{∗})_{tors}

has finite exponent.

The cokernel in the previous theorem can be infinite. See Example 6.4 below.

Corollary 6.2. Let E_{i}, i = 1, …, r be intermediate fields for the finite separable extension K/k. Let ℓ be prime. If

(i) ℓ is odd and μ_{ℓ} ⊂ k, (ii) ℓ = 2 and μ_{4} ⊂ k, or (iii) the ℓ-primary subgroup of μ(K) is finite,

then the ℓ-primary torsion subgroup of K^{∗}/E_{1}^{∗} … E_{r}^{∗} has bounded exponent. In particular, this holds if k is finitely generated over the prime field.

Proof. This follows from Lemma 4.2 and Theorem 6.1.

Let E_{i}, i = 1, …, r be intermediate fields for the finite separable extension K/k. Let L be the Galois closure of K/k with L/k of degree n. Then L embeds into L ⊗_{k} L ≅ L^{n} via y → 1 ⊗ y. Indeed, we can view L ⊗_{k} L as a G-module where G acts trivially on the first factor and naturally on the second factor. Let f : L ⊗_{k} L → L ⊗ ZG be defined by f(x ⊗ y) = ∑_{g∈G} g(x)y ⊗ g. Note that f is a G-module map (where G acts diagonally on the righthand side). Moreover, L ⊗ ZG has an algebra structure (coordinatewise) and that f is an algebra isomorphism. Thus f induces an isomorphism of (L ⊗ L)^{∗} with L^{∗} ⊗ ZG with the diagonal action.

Restrict f to L^{∗} (with the nontrivial action of G). So f injects L^{∗} into W := L^{∗} ⊗ ZG via y → y ⊗ N_{G}, where N_{g} = ∑_{g∈G} g. This induces a homomorphism of abelian groups

f̄: L^{∗}/E_{1}^{∗} … E_{r}^{∗} → W/(W^{C_{1}} + … + W^{C_{r}}).

If r = 2, then recall (Theorem 2.6) that in sequence (4), S is a permutation lattice and T is torsionfree. It follows easily from (10′) for both L^{∗} and W and the fact that H^{1}(G, L^{∗}⊗S) = H^{1}(G, W ⊗ S) = 0 that the kernel of f̄ is 0.

If r > 2, we can apply Lemma 2.11. Thus, we have:

Proposition 6.3. The map f̄ defined above is an injection if r ≤ 2. In general, the kernel of f̄ has finite exponent dividing |G|.

We now consider a modified version of the [HW] approach. Let K/k be a finite separable extension of fields with proper intermediate fields E_{1} and E_{2}. Let P be a maximal ideal of B = E_{2}[X] with B/P ≅ K. Let A be the subring of B containing P with A/P ≅ E_{1}. Then B is the normalization of A and P is the conductor of A. Since Pic(B) is trivial, it follows that Pic(A) = D(A) ≅ K^{∗}/E_{1}^{∗}E_{2}^{∗}. Thus results about Picard groups can be applied to this group.

We may assume that k is a maximal subfield of A. Let L be the Galois closure of K/k and G the Galois group of L/k. Set R = A ⊗_{k} L. Then

Pic(R) ≅ (K ⊗_{k} L)^{∗}/(E_{1} ⊗_{k} L)^{∗}(E_{2} ⊗_{k} L)^{∗}.

As in §6, we can identify these terms as the fixed points of certain subgroups of G where G is acting on (L ⊗_{k} L)^{∗}. This last module is isomorphic to L^{∗} ⊗ ZG as a G-module. Thus, by Proposition 2.5(b), Pic(R) is isomorphic to a finite direct sum of copies of L^{∗}. In particular, the n-torsion subgroup of Pic(R) is finite for all n ≥ 1.

Let J be the kernel of the map from Pic(A) to Pic(R). It is straightforward to compute that J has exponent dividing |G|. If the order of G is not divisible by the characteristic of k, then [W, Theorem 2.1] applies and the kernel is finite. More generally, if m is not a multiple of the characteristic of k, [W, Theorem 2.1] implies that the m-torsion subgroup of J is finite. Thus, if m is not a multiple of the characteristic of k, the m-torsion subgroup of Pic(A) is finite.

However, we have seen (Proposition 6.3) that in fact J is trivial. This already yields an improvement of [HW] – showing that there is no p-torsion in Pic(A) ≅ K^{∗}/E_{1}^{∗}E_{2}^{∗} and that the m-torsion subgroup is finite for all m ≥ 1.

The finiteness of the kernel of f̄ also is closely related to the question of whether or not S in the sequence (4) is coflasque. This is particularly evident for finitely generated fields (Theorem 7.2).

We now produce an explicit example with the kernel of f̄ finite but nontrivial.

Example 6.4. Let L/k be a Galois extension of degree 4 with Galois group G = Z/2×Z/2. Let E_{i}, 1 ≤ i ≤ 3, be the three quadratic subfields. Let N be the norm map from L to k. Let J = ker(f̄).

(a) Then J maps onto N(L^{∗}) ∩ L^{∗}^{2}/k^{∗}^{2}. In particular, J may be nontrivial.

(b) If k is finitely generated over the prime field, then L^{∗}/E_{1}^{∗}E_{2}^{∗}E_{3}^{∗} is an infinite group of exponent 2 and J is finite.

Proof. Let N_{i} be the norm map from E_{i} to k. If x ∈ L^{∗}, then f(x) = x ⊗ N_{G}. We can identify W with L^{∗} ⊗ ZG where G acts trivially on L^{∗} via x ⊗ g → g^{−1}(x) ⊗ g. With this identification, f(x) = ∑_{g} g^{−1}(x) ⊗ g. If h ∈ G, then

W^{h} = {∑_{g} a_{g} ⊗ g : a_{hg} = a_{g}},

and so the product of coordinates of any element of λ(W) is a square in L. Conversely, it is straightforward to see that if w ∈ W has that property, then w ∈ λ(W).

Thus f(x) ∈ λ(W) if and only if N(x) is a square in L. On the other hand, if x ∈ E_{1}^{∗}E_{2}^{∗}E_{3}^{∗}, then, obviously, N(x) is a square in k.

Let R be the subgroup of L^{∗} consisting of those elements whose norm is a square in L. The previous paragraph shows that J ≅ R/E_{1}^{∗}E_{2}^{∗}E_{3}^{∗} and that N(λ(L^{∗})) ⊆ k^{∗}^{2}. This proves the first assertion of (a).

Consider the special case k = Q and L = Q(i, \sqrt{2}). If x = 1 + i + \sqrt{2}, N(x) = 8 is a square in L but not in k. Thus J is nontrivial. This completes the proof of (a).

(b) If x ∈ L^{∗}, then x^{2}N(x) = ∏_{1≠g∈G} xg(x) ∈ E_{1}^{∗}E_{2}^{∗}E_{3}^{∗} and so L^{∗}/E_{1}^{∗}E_{2}^{∗}E_{3}^{∗} is a group of exponent 2 and so certainly the kernel is as well. If L is finitely generated over the prime field, then the methods of the next section show that L^{∗} is a direct sum of a finitely generated G-module and permutation lattices. It follows from Proposition 2.10 that in the case under discussion S (as in (4)) is a rank-two trivial G-module. It is straightforward (using the methods of section 7, the structure of S and sequence (10′) for M = L^{∗} and M = W) to show that the kernel is finite.

We note that k is not an algebraic extension of a finite field since G is not cyclic. It follows from [GW, (1.5)(c), (1.6), (1.8)] that L^{∗}/E_{1}^{∗}E_{2}^{∗}E_{3}^{∗} is an infinite group of exponent 2.

Using Example 2.9 and the methods of section 7, we can produce fields where ker(f̄) is infinite.

Example 6.5. Let L/k be a Galois extension of number fields with Galois group G elementary abelian of order 8. Let E_{i}, 1 ≤ i ≤ 3 be intermediate fields with [L : E_{i}] = 2 and k = E_{1} ∩ E_{2} ∩ E_{3}. Then ker(f̄) is infinite.

Proof. As above, f̄ maps L^{∗}/E_{1}^{∗}E_{2}^{∗}E_{3}^{∗} to L^{∗} ⊗ ZG/λ(L^{∗} ⊗ ZG). It is straightforward from the methods of section 2 to show that the kernel of the map Z[G/H]/λ(Z[G/H]) → (Z[G/H] ⊗ Z[G])/λ(Z[G/H] ⊗ Z[G]) is isomorphic to H^{1}(H, S). By Example 2.9, there is a cyclic subgroup C of G with H^{1}(C, S) nonzero. Since Z[G/C] occurs as a summand of L^{∗} infinitely often (see §7), it follows that ker(f̄) is infinite.

We next give an example with K^{∗}/E_{1}^{∗}E_{2}^{∗}E_{3}^{∗} finite but nontrivial. It follows from §7 that if K is finitely generated over the prime field, then the quotient being finitely generated implies that T = 0, whence the quotient is also trivial. Al Sethuraman had suggested considering fields which are the intersection of a finite number of Henselizations.

Example 6.6. Let q be a prime. There exists a field k of arbitrary characteristic p ≠ q and a Galois extension K/k with Galois group Z/q × Z/q such that if E_{1}, …, E_{r} are all the intermediate extensions of K/k, then K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} is finite but nontrivial.

Proof. Let F be an algebraically closed field of characteristic p for any p ≠ q. Let g_{i} be three distinct monic linear polynomials in F [x]. Let v_{i} be the corresponding valuation. Let F_{i} be a Henselization of F (x) with respect to v_{i}. We also let v_{i} denote the extension of v_{i} to F_{i}. Set k = F_{1} ∩ F_{2} ∩ F_{3}. Let h_{i} be a q-th root of g_{i}. Finally, set K = F [h_{2}, h_{3}]. Since v_{i}(g_{j}) = δ_{ij}, it follows that K/k is a Galois extension with Galois group G ≅ Z/q × Z/q.

Also, since the residue field of the valuation ring of F_{i} is algebraically closed, it follows that any element t of F_{i} with v_{i}(t) divisible by q is a qth power in F_{i}. Hence t ∈ F is a qth power if and only v_{i}(t) is divisible by q for each i. In particular k^{∗}/(k∗)^{q} has order q^{3}. It follows that K^{∗}/(K^{∗})^{q} is finite (this property is inherited by cyclic extensions of degree q – see [La, Cor. 3.5, p. 203] for the case q = 2 – the proof given just above the Corollary is valid for any prime).

Let E_{i}, 1 ≤ i ≤ r be the set of extensions of k of degree q contained in K. It follows by [GW 1.6] that (K^{∗})^{q} ⊆ E_{1}^{∗} ⋯ E_{r}^{∗} and so K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} is finite.

Note that v_{1} is unramfied in K (since v_{1}(g_{2}) = v_{1}(g_{3}) = 0). Since the residue field is algebraically closed, this implies that v_{1} splits completely from k to K. This yields a G-module surjection from K^{∗} to ZG (defined by x ↦ (w_{1}(x), …, w_{4}(x)) where the w_{i} are the extensions of v_{1}). It follows from [GW, 1.6] that K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} is nontrivial (or from the observation that T has order q for any noncyclic abelian q-group).

We close this section with some remarks about the torsionfree rank of K^{∗}/E_{1}^{∗} … E_{r}^{∗}. These results are already in [GW, Theorem 1.5]. Since there is no harm in replacing E_{i} by its purely inseparable closure in K, we may assume that K is separable over E_{i}.

Theorem 6.7. Let K/k be a finite separable extension of fields. Let E_{1}, … E_{r} be proper intermediate subfields with k = E_{1} ∩ … ∩ E_{r}. Assume that K is not algebraic over a finite field. Then Q ⊗ (K^{∗}/E_{1}^{∗} … E_{r}^{∗}) = 0 if T is finite and has infinite dimension otherwise. In particular, if r ≤ 2, (K^{∗}/E_{1}^{∗} … E_{r}^{∗}) has infinite torsionfree rank.

Proof. We apply Proposition 2.15 with M = L^{∗}, L the Galois closure of K/k. The key observation is that Q ⊗ L^{∗} has a free QG submodule of infinite rank (see [GW, proof of 1.5]; indeed the main result of [Ji] is that Q ⊗ L^{∗} is a free QG-module of rank equal to the cardinality of the field). If r ≤ 2, then T is torsionfree of positive rank (by Theorem 2.6), hence infinite.

7. Finitely Generated Fields

If k is finitely generated over the prime field, one can take a different approach. For the rest of this section, assume that k is such a field. Assume also that k is infinite. As usual K/k is a finite separable extension. Let L be the Galois closure.

Choose a domain A finitely generated over Z and with quotient field L. If E is an intermediate field, set A_{E} = A ∩ E. Let E_{i}, i = 1, …, r be a collection of intermediate fields and set A_{i} = A ∩ E_{i}. By inverting an element of A, we may assume that all A_{i} are regular. Classical and deep results due to Mordell, Weil, Severi, Néron imply (Roquette, see [L], Thm. 6.6.2] that Pic(A^{C}) is finitely generated for every subgroup C of G. So by inverting another element of A if necessary, we can assume that Pic(A^{C}) = 0 for each subgroup C and that A/(A ∩ k) is unramified (i.e. the discriminant is a unit).

Then (since Pic(A) = 0) we have the sequence

0 → A^{∗} → L^{∗} → Γ → 0

where Γ is a direct sum of permutation lattices (corresponding to discrete rank-one valuations of L centered on height-one primes of A).

Since Pic(A^{C}) = 0 and the extension A/A^{C} is unramified, H^{1}(C, A^{∗}) = 0 for any subgroup C. Since Γ is a direct sum of permutation lattices, this implies that Ext_{G}^{1}(Γ, A^{∗}) = 0 and the sequence above is split exact as a sequence of G-modules. Thus

K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} ≅ (A ∩ K)^{∗}/A_{1}^{∗} ⋯ A_{r}^{∗} ⊕ Γ^{D}/(Γ^{C_{1}} + … + Γ^{C_{r}}),

where C_{i} is the subgroup of G acting trivially on E_{i}.

Now assume that r ≤ 2. Since Γ is a direct sum of permutation lattices, Proposition 2.5 implies that the torsion subgroup of L^{∗}/E_{1}^{∗}E_{2}^{∗} is isomorphic to the torsion subgroup of A^{∗}/A_{1}^{∗}A_{2}^{∗}. Since A^{∗} is finitely generated (see [L, 7.2, p. 42]) we have proved a weakened version of Corollary 5.3:

Theorem 7.1. If k is a field finitely generated over the prime field, K is a finite separable extension field of k and E_{1} and E_{2} are proper subfields of K/k, then K^{∗}/E_{1}^{∗}E_{2}^{∗} is a direct sum of a countably generated free abelian group and a finite group. In particular, the torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} is finite.

Note that this proof does not show that the p-torsion subgroup of K^{∗}/E_{1}^{∗}E_{2}^{∗} is trivial if k has positive characteristic p.

For general r, we see that L^{∗}/λ(L) ≅ A^{∗}/λ(A)⊕Γ/λ(Γ). The left-hand term is finitely generated. The right-hand term essentially depends only on G and the C_{i} (and of course, the number of transitive permutation lattices of each type which are summands of Γ).

Theorem 7.2. Let k be a field which is finitely generated over the prime field but is not algebraic over a finite field, K a finite separable extension field of k and E_{1}, …, E_{r} proper subfields of K containing k. Let L be the Galois closure of K/k and G the Galois group of L/k. Let C_{i} be the Galois group of L/E_{i}. Define S and T as in (4). Then:

(a) If T is not torsionfree, K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} has infinite torsion subgroup (of finite exponent).

(b) If T is torsionfree, then (K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗})_{tors} ≅ Y ⊕ H^{1}(G, Γ ⊗ S), where Y is finite.

(c) If T is torsionfree and k is a number field, then

(K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗})_{tors} ≅ ⊕_{i=1}^{∞}(⊕_{C∈S}H^{1}(C, S)) ⊕ Y,

where Y is finite and S is the collection of all cyclic subgroups of G.

(d) If T is torsionfree, k = k_{0}(x), and L = L_{0}(x) where k_{0} is a subfield of k, then

(K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗})_{tors} ≅ ⊕_{i=1}^{∞}(⊕_{C∈S}H^{1}(C, S)) ⊕ Y,

where Y is finite and S is the collection of all subgroups of G.

Proof. By [GW, 1.8 and proof of 1.5], Γ has a ZG-free direct summand of infinite rank. Thus Corollary 2.3(f) implies that K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} has a direct summand isomorphic to an infinite direct sum of copies of T. In particular, if T is not torsionfree, the torsion subgroup of K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} is infinite. It has finite exponent by Theorem 6.1. This proves (a). Since Γ is coflasque and torsionfree, (b) follows from Corollary 2.3(e).

If k is a number field, then since all residue fields are finite, the decomposition groups are cyclic (the extension A/A^{G} is unramified by the choice of A). The Chebotarev density theorem implies that each cyclic subgroup occurs as a decomposition group for infinitely many primes. Thus

Γ ≅ ⊕_{i=1}^{∞}(⊕_{C∈S}Z[G/C]).

Corollary 2.3(f) implies (c).

(d) Under these hypotheses it follows that every subgroup of G will occur as a decomposition group infinitely often (e.g., consider the primes x − a for various a in L). Now argue as in (c).

The next result is an immediate consequence of Theorem 7.2.

Corollary 7.3. Let G be a finite group with subgroups C_{1}, …, C_{r} with D ⊆ ∩_{i}C_{i}. Let S and T be defined by (4). The following are equivalent:

(a) There exists no finite separable extension of fields K/k finitely generated over the prime field such that the Galois closure L/k has Galois group G and K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} has an infinite torsion group, where E_{i} is the fixed field of C_{i}.

(b) T is torsionfree and S is coflasque.

The previous theorem allows us to give many examples in which T is torsion and so the torsion subgroup of K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} infinite (see also [GW, 1.4]). In particular, if G is a noncyclic p-group and C_{1}, …, C_{r} are the subgroups of index p, T will be an elementary abelian p-group (it is also easy to see in this case that S is a trivial G-module and so is coflasque).

Example 2.9 (see also 6.5) also shows that there are examples of number fields such that T is torsionfree and the torsion subgroup of K^{∗}/E_{1}^{∗} ⋯ E_{r}^{∗} is infinite.

The proof of 7.1 extends to the case when K is separable or semisimple (using the ideas in 5.5 and 5.6). Thus one has:

Theorem 7.4. Let k be a field finitely generated over the prime field and K a finite dimensional semisimple commutative k-algebra. Let E_{1} and E_{2} be intermediate subalgebras of K/k.

(a) If k has positive characteristic p, then K^{∗}/E_{1}^{∗}E_{2}^{∗} is a direct sum of a countably generated free abelian group, a finite group and a p-group of finite exponent.

(b) If K/k is separable, then K^{∗}/E_{1}^{∗}E_{2}^{∗} is a direct sum of a countably generated free abelian group and a finite group.

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J.-L. Colliot-Thélène  Robert M. Guralnick
Mathématiques, Bât. 425,  Department of Mathematics
CNRS, Université de Paris-Sud  University of Southern California
F-91405 Orsay, France  Los Angeles, CA 90089-1113 USA
e-mail: colliot@matups.matups.fr  e-mail: guralnic@mtha.usc.edu

Roger Wiegand Department of Mathematics University of Nebraska Lincoln, NE 68588-0323 USA e-mail: rwiegand@unlinfo.unl.edu