curly braces 1

Kummer subfields of tame division algebras over Henselian

valued fields^{∗}^{†}

Karim Mounirh

Département des Mathématiques Faculté des Sciences Université Catholique de Louvain Belgium e.mail: akamounirh@hotmail.com

Abstract : By generalizing the method used by Tignol and Amitsur in [TA85], we determine necessary and sufficient conditions for an arbitrary tame central division algebra D over a Henselian valued field E to have Kummer subfields [Corollary 2.11 and Corollary 2.12]. We prove also that if D is a tame semiramified division algebra of prime power degree p^{n} over E such that p ≠ char(Ē) and rk(Γ_{D}/Γ_{F}) ≥ 3 [resp., such that p ≠ char(Ē) and p^{3} divides exp(Γ_{D}/Γ_{E})], then D is non-cyclic [Proposition 3.1] [resp., D is not an elementary abelian crossed product [Proposition 3.2]].

Introduction

Let B be a tame central division algebra over a Henselian valued field E. We know by [JW90, Lemma 6.2] that B is similar to some S ⊗_{E} T, where S is an inertially split [resp., T is a tame totally ramified] division algebra over E. By generalizing the method used by Tignol and Amitsur in [TA85], Morandi and Sethuraman determined in [MorSe95] necessary and sufficient conditions for B to have Kummer subfields when B = S ⊗_{E} T. A good question was to see if we have the same results when B is an arbitrary tame central division algebra over E. To deal with this question, we remarked that it will be the same if we can determine necessary and sufficient conditions for a graded central division algebra over a graded field to have Kummer graded subfields. Indeed, we know that if char(Ē) does not divide deg(B), then any result concerning graded subfields of GB gives an analogous one for B. A first key idea was the fact that if D is a graded central division algebra over a graded field F, then there is a factor set (ω, f) of Γ_{D}/Γ_{F} in D_{0}F such that D is the generalized graded crossed product (D_{0}F, Γ_{D}/Γ_{F} , (ω, f)). Another important result consists in the fact that f can be decompsed in a nice way. Indeed, we showed that for any γ̄, γ̄′ ∈ Γ_{D}/Γ_{F}, we can write f(γ̄, γ̄′) = d(γ̄, γ̄′)h(γ̄, γ̄′), where (ω, d) is a factor set of Γ_{D}/Γ_{F} in D_{0} and h ∈ Z^{2}(Γ_{D}/Γ_{F} , F^{∗})_{sym} [Lemma 1.6]. We show also in section 2 that if K is a Kummer graded subfield of D, then there is an exact sequence of trivial Γ_{K}/Γ_{F} -modules α_{K} : 1 → kum(K_{0}/F_{0}) → kum(K/F) → Γ_{K}/Γ_{F} → 0. We consider α_{K} as an element of Z^{2}(Γ_{D}/Γ_{F} , kum(K_{0}/F_{0}))_{sym} and so applying the previous facts we get in [Corollary 2.10 and Corollary 2.11] necessary and sufficient conditions for D to have Kummer graded subfields when F_{0} contains enough roots of unity. This results are then applied to give necessary and sufficient conditions for a semiramified graded division algebra D over a graded field F to be cyclic [resp., to be an elementary abelian graded crossed product] when F_{0} contains enough roots of unity. In section 3, and without assuming any root of unity to be in Ē, we prove that if E is a Henselian valued field and B is a tame semiramified division algebra of prime power degree p^{n} over E such that p ≠ char(Ē) and rk(Γ_{B}/Γ_{F}) ≥ 3 [resp., such that p ≠ char(Ē) and p^{3} divides exp(Γ_{B}/Γ_{E})], then B is non-cyclic [Proposition 3.1] [resp., B is not an elementary abelian crossed product [Proposition 3.2]].

∗Mathematics subject classification (MSC2000): 16K50, 16W50, 16W60 and 16W70.

†Key Words: Generalized crossed products, (Graded) Brauer group, Valued division algebras, Henselization, Graded division algebras, Kummer graded field extensions.

Throughout this paper, we assume familiarity with the definitions and notations previously used in [M05] and [M07].

1 Generalized graded crossed products and graded division algebras

(1.1) Let L be a field and A a central simple algebra over L. We denote by A^{∗} the group of invertible elements of A and by Aut(A) the group of ring automorphisms of A. For any c ∈ A^{∗}, we denote by Inn(c) the ring automorphism of A defined by a ↦ cac^{−1}. Let H be a finite group that acts by automorphisms on L and let ω : H → Aut(A) and f : H × H → A^{∗} be two maps. We say that (ω, f) is a factor set of H in A if the following conditions are satisfied :

(1) ω_{σ}(a) = σ(a) for all a ∈ L and σ ∈ H,

(2) ω_{σ}ω_{τ} = Inn(f(σ, τ))ω_{στ} for all σ, τ ∈ H, and

(3) f(σ, τ)f(στ, μ) = ω_{σ}(f(τ, μ))f(σ, τμ) for all σ, τ, μ ∈ H. If (ω, f) is a factor set of H in A, then we define the generalized crossed product associated to (ω, f) to be the algebra (A, H, (ω, f)) = ⊕_{σ∈H}Ax_{σ}, where x_{σ} are independent indeterminates over A satisfying the following multiplicative conditions (for all σ ∈ H and a ∈ A) :

(4) x_{σ}a = ω_{σ}(a)x_{σ}, and

(5) x_{σ}x_{τ} = f(σ, τ)x_{στ}.

It is well-known that if char(L) does not divide card(H), then (A, H, (ω, f)) is a semisimple algebra (see [MorSe95, p. 556]).

Let (ω, f) and (ω′, f′) be two factor sets of H in A. We say that (ω, f) and (ω′, f′) are cohomologous if there is a family (a_{σ})_{σ∈H} of elements of A^{∗} such that for all σ, τ ∈ H, ω′_{σ} = Inn(a_{σ})ω_{σ} and f′(σ, τ) = a_{σ}ω_{σ}(a_{τ})f(σ, τ)a_{στ}^{−1}. We write in this case (ω, f) ∼ (ω′, f′). The relation ∼ is an equivalence relation on the set of factor sets of H in A. We denote the set of equivalence classes by H(H, A^{∗}). If A = L is a Galois field extension of some field E and H = Gal(L/E), then H(H, A^{∗}) is the second Galois cohomology group H^{2}(H, L^{∗}).

Now, let L be a graded field, A a graded central simple algebra over L, H a finite group that acts on L by graded automorphisms (of grade 0), GAut(A)_{0} the group of graded ring automorphisms (of grade 0) of A (i.e. ring automorphisms of A such that f(A_{δ}) = A_{δ}). In the same way as above, if ω : H → GAut(A)_{0} and f : H × H → A^{∗} are two maps that satisfy the conditions (1) to (3) above, then we say that (ω, f) is a graded factor set of H in A. The corresponding graded generalized crossed product (A, H, (ω, f)) is defined also as above. Namely, (A, H, (ω, f)) = ⊕_{σ∈H}Ax_{σ}, where x_{σ} are independent indeterminates on A satisfying the multiplicative conditions : x_{σ}a = ω_{σ}(a)x_{σ} and x_{σ}x_{τ} = f(σ, τ)x_{στ} for all a ∈ A and σ, τ ∈ H. As we will see in the next lemma, (A, H, (ω, f)) has a unique graded algebra structure extending that of A and for which x_{σ} are homogeneous elements (the proof of this lemma is inspired from [HW(2), Lemma 5.4]).

Lemma 1. 2 Let L be a graded field, A be a graded central simple algebra over L, H a finite group that acts on L by graded automorphisms, and (ω, f) a graded factor set of H in A. Then, there is a unique graded algebra structure of (A, H, (ω, f)) extending the grading of A and for which x_{σ} are homogeneous elements.

Proof. Let Γ_{A} (a totally ordered abelian group) be the support of A, Δ_{A}(= Γ_{A} ⊗_{ZZ} Q ′ ) be the divisible hull of Γ_{A} and consider the map h : H × H → Δ_{A}, (σ, τ) ↦ gr(f(σ, τ)). Then, it follows from condition (3) above that h is a cocycle of Z^{2}(H, Δ_{A}) (for the trivial action of H on Δ_{A}). Since H is finite and Δ_{A} is uniquely divisible, then H^{2}(H, Δ_{A}) = H^{1}(H, Δ_{A}) = 0. Therefore, there is a unique family (δ_{σ})_{σ∈H} of elements of Δ_{A} such that h(σ, τ) = δ_{σ} + δ_{τ} − δ_{στ} (the uniqueness follows from the fact that H^{1}(H, Δ_{A}) = 0). The unique graded structure of (A, H, (ω, f)) that extends that of A and for which x_{σ} are homogeneous elements is then defined by gr(x_{σ}) = δ_{σ}.

In what follows, we will show that any graded division algebra can be represented as a generalized graded crossed product. This representation, will be applied in section 2 to determine necessary and sufficient conditions for the existence of Kummer graded subfields.

(1.3) Let F be a graded field and D a graded central division algebra over F. Then, the map θ_{D} : Γ_{D}/Γ_{F} → Gal(Z(D_{0})/F_{0}), defined by θ_{D}(gr(d) + Γ_{F})(a) = dad^{−1} for any d ∈ D^{∗} and a ∈ Z(D_{0}), is a surjective group homomorphism. Since HCq(D) is a tame central division algebra over HF rac(F), then by [JW90, Proposition 1.7 and Definition p. 166] Z(D_{0}) is an abelian field extension of F_{0}. For simplicity, we denote by G the Galois group Gal(Z(D_{0})/F_{0}). So, by [HW(1)99, Remark 3.1] Z(D_{0})F is an abelian Galois graded field extension of F with Galois group isomorphic to G. In what follows, we will consider the action of Γ_{D}/Γ_{F} on Z(D_{0})F defined by θ_{D} (i.e., for any γ̄ ∈ Γ_{D}/Γ_{F} and any a ∈ Z(D_{0})F, we let γ̄(a) = dγ̄ad−1 ; γ̄, where dγ̄ is an arbitrary homogeneous element of D^{∗} such that gr(dγ̄) + Γ_{F} = γ̄). We aim here to show that there is a graded factor set (ω, f) of H := Γ_{D}/Γ_{F} in D_{0}F such that D = (D_{0}F, H, (ω, f)). For this, we fix a family of homogeneous elements (zγ̄)_{γ̄∈H} of D^{∗} with gr(zγ̄) + Γ_{F} = γ̄. Clearly, we have D = ⊕_{γ̄∈H}D_{0}F zγ̄ (because both graded algebras have the same 0-component and the same support). We define :

ω : H → GAut(D_{0}F)_{0}

and

f : H × H → (D_{0}F)^{∗}

by ωγ̄(a) = zγ̄az−1 ; γ̄ and f(γ̄, γ̄′) = zγ̄z_{}_{γ̄′}z_{γ̄+γ̄′}^{−1}. One can easily see that (ω, f) is a graded factor set of H in D_{0}F. So, D = ⊕_{γ̄∈H}D_{0}F zγ̄ = (D_{0}F, H, (ω, f))

Let B = ⊕_{γ̄∈ker(θ_{D})}D_{0}F zγ̄ and for any σ ∈ G choose a γ̄_{σ} ∈ H such that θ_{D}(γ̄_{σ}) = σ and let z_{σ} := zγ̄_{σ}. Then, we have the following Proposition.

Proposition 1. 4 B is the centralizer of Z(D_{0}F) in D and D = ⊕_{σ∈G}Bz_{σ} = (B, G, (w, g)) for some graded factor set (w, g) of G in B.

Proof. Let C be the centralizer of Z(D_{0})F in D. Clearly, we have B ⊆ C. Moreover, by [HW(2)99, Proposition 1.5] we have [C : F] = [D : F]/[Z(D_{0})F : F] = [D_{0} : F_{0}](Γ_{D} : Γ_{F})/[Z(D_{0}) : F_{0}] = [D_{0} : F_{0}]|ker(θ_{D})| = [B : F]. Hence, B = C. Clearly, we have ⊕_{σ∈G}Bz_{σ} = ⊕_{σ∈G}(⊕_{γ̄∈ker(θ_{D})}D_{0}F zγ̄)z_{σ} = ⊕_{γ̄∈Γ_{D}/Γ_{F}} D_{0}F zγ̄ = D. Let

w : G → GAut(B)_{0}

and

g : G × G → B^{∗}

be the maps defined by w_{σ}(b) = z_{σ}bz_{σ}^{−1} (for any b ∈ B and σ ∈ G) and g(σ, τ) = z_{σ}z_{τ}z_{στ}^{−1} (for any σ, τ ∈ G). Then, (w, g) is a graded factor set of G in B and (B, G, (w, g)) = ⊕_{σ∈G}Bz_{σ} = D.

Remark 1.5 Remark that the existence of (w, g) in Lemma 1.4 follows also by the graded version of [T87, Theorem 1.3(b)].

(1.6) Now, with the notations of (1.3) let S = (δ̄_{i} := δ_{i} + Γ_{F})_{1≤i≤r} a basis of H, q_{i} = ord(δ̄_{i}) for 1 ≤ i ≤ r and I = {(m_{1}, ..., m_{r}) ∈ IN^{r} | 0 ≤ m_{i} < q_{i} for 1 ≤ i ≤ r}. We fix a family (x_{i})_{1≤i≤r} of elements of F^{∗} with gr(x_{i}) = q_{i}δ_{i}, and we consider a family (z_{i})_{1≤i≤r} of elements of D^{∗} with gr(z_{i}) = δ_{i}. For m̄ = (m_{1}, ..., m_{r}) ∈ I, we let m̄δ̄ = ∑_{1≤i≤r} m_{i}δ̄_{i} and zm̄ = ∏_{i=1}^{r} z_{i}^{m_{i}}. Remark that for any γ̄ ∈ H, there is a unique element m̄ ∈ I such that γ̄ = m̄δ̄. Henceforth, for any γ̄ = m̄δ̄ (where m̄ ∈ I), we choose zγ̄ = zm̄. Let f : H × H → (D_{0}F)^{∗} be the map previously defined in (1.3) by f(γ̄, γ̄′) = zγ̄z_{}_{γ̄′}z_{γ̄+γ̄′}^{−1}. Then, for any m̄, n̄ ∈ I, f(m̄δ̄, n̄δ̄) = zm̄zn̄z^{−β(m̄+n̄)}, where β(m̄+n̄) ∈ I with m̄+n̄ ≡ β(m̄+n̄) mod ∏_{i=1}^{r} q_{i}ZZ. Write m_{i} +n_{i} = β(m̄+n̄)_{i} +t_{i}q_{i}, where t_{i} ∈ IN, then f(m̄δ̄, n̄δ̄) = d(m̄δ̄, n̄δ̄)h(m̄δ̄, n̄δ̄), where d(m̄δ̄, n̄δ̄) ∈ D_{0}^{∗} and h(m̄δ̄, n̄δ̄) = ∏_{i=1}^{r} x_{i}^{t_{i}}. Consider the map ω defined in (1.3), we will denote also by ω the map : H → Aut(D_{0}) defined by γ̄ ↦ ωγ̄_{/D_{0}}. We have the following lemma.

Lemma 1. 7 (ω, d) is a factor set of H in D_{0} and h ∈ Z^{2}(H, F^{∗})_{sym}.

Proof. Let m̄, n̄ and s̄ be elements of I. Since H acts trivially on F^{∗}, then

m̄δ̄h(n̄δ̄, s̄δ̄)h(m̄δ̄, n̄δ̄ + s̄δ̄) = h(n̄δ̄, s̄δ̄)h(m̄δ̄, β(n̄ + s̄)δ̄) = (∏_{i=1}^{r} x_{i}^{λ_{i}})(∏_{i=1}^{r} x_{i}^{γ_{i}})

where λ_{i} = \frac{1}{q_{i}} (n_{i} + s_{i} − β(n̄ + s̄)_{i}) and γ_{i} = \frac{1}{q_{i}} (m_{i} + β(n̄ + s̄)_{i} − β(m̄ + β(n̄ + s̄))_{i}). We have β(m̄ + β(n̄ + s̄)) = β(m̄ + n̄ + s̄), hence

m̄δ̄h(n̄δ̄, s̄δ̄)h(m̄δ̄, n̄δ̄ + s̄δ̄) = (∏_{i=1}^{r} x_{i}^{ξ_{i}}).

where ξ_{i} = \frac{1}{q_{i}} m_{i} + n_{i} + s_{i} − β(m̄ + n̄ + s̄)_{i}.

Likewise, we have :

h(m̄δ̄, n̄δ̄)h(m̄δ̄ + n̄δ̄, s̄δ̄) = ∏_{i=1}^{r} x_{i}^{ξ_{i}}.

Moreover, it is clear that h(m̄δ̄, n̄δ̄) = h(n̄δ̄, m̄δ̄). Hence, h ∈ Z^{2}(H, F^{∗})_{sym}. The fact that (ω, f) is a graded factor set of H in D_{0}F and that h ∈ Z^{2}(H, F^{∗})_{sym} imply (ω, d) is a factor set of H in D_{0}.

Remark 1.8 If D is a semiramified graded division algebra over F, then using the same arguments as in the proof of Lemma 1.7, we prove that d ∈ Z^{2}(H, D_{0}^{∗}) (see that in this case H ≅ Gal(D_{0}/F_{0})).

2 Kummer graded subfields of graded division algebras

(2.1) Let F be a graded field and K is a finite-dimensional abelian graded field extension of F (i.e., such that F rac(K)/F rac(F) is an abelian Galois field extension [see HW(1)99]). We say that K is a Kummer graded field extension of F if F_{0} contains a primitive m^{th} root of unity, where m is the exponent of Gal(K/F). In such a case, as for ungraded Kummer field extensions, we set KUM(K/F) = {x ∈ K^{∗} | x^{m} ∈ F } and kum(K/F) = KUM(K/F)/F^{∗}. One can easily see that kum(K/F) is isomorphic to Gal(K/F). Now, let K be a Kummer graded field extension of F, then we have K = F [a | a ∈ KUM(K/F)], so Γ_{K}/Γ_{F} is generated by {gr(a) + Γ_{F} | a ∈ KUM(K/F)}, therefore the group homomorphism ψ : kum(K/F) → Γ_{K}/Γ_{F}, defined by ψ(aF^{∗}) = gr(a)+Γ_{F}, for a ∈ KUM(K/F), is surjective. Let ϕ : kum(K_{0}/F_{0}) → kum(K/F) be the group homomorphism defined by ϕ(aF_{0}^{∗}) = aF^{∗}, for every a ∈ KUM(K_{0}/F_{0}). Clearly, ϕ is injective and ψ ∘ ϕ = 0. By comparing the cardinalities, we conclude that the following sequence of trivial Γ_{K}/Γ_{F} -modules :

α_{K} : 1 → kum(K_{0}/F_{0}) →^{ϕ} kum(K/F) →^{ψ} Γ_{K}/Γ_{F} → 0

is exact. Remark that since kum(K/F) is abelian, then α_{K} ∈ Z^{2}(Γ_{K}/Γ_{F} , kum(K_{0}/F_{0}))_{sym}.

(2.2) With the notations of (2.1), we have KUM(K/F) ∩ D_{0} = KUM(K_{0}/F_{0}). Indeed, let a ∈ KUM(K/F) ∩ D_{0}, then ψ(aF^{∗}) = 0, so aF^{∗} ∈ im(ϕ). Hence there is b ∈ KUM(K_{0}/F_{0}) such that aF^{∗} = bF^{∗}. Since both a and b are in D_{0}^{∗}, then ab^{−1} ∈ F_{0}^{∗}(= D_{0}^{∗} ∩F^{∗}). So, a ∈ KUM(K_{0}/F_{0}). This shows that KUM(K/F)∩D_{0} ⊆ KUM(K_{0}/F_{0}). The converse inclusion is trivial.

2.3 Notations : We precise here some notations needed for the next result :

(a) Let e : KUM(K_{0}/F_{0}) → kum(K_{0}/F_{0}) be the canonical surjective homomorphism. We denote by e_{∗} : H^{2}(Γ_{K}/Γ_{F} , KUM(K_{0}/F_{0}))_{sym} → H^{2}(Γ_{K}/Γ_{F} , kum(K_{0}/F_{0}))_{sym} the corresponding homomorphism of cohomology groups (for the trivial action of Γ_{K}/Γ_{F} on KUM(K_{0}/F_{0}) and on kum(K_{0}/F_{0})).

(b) Let (ω, d) be the factor set of H in D_{0} previously seen in Lemma 1.7, we denote by res_{Γ_{K}/Γ_{F}}^{H} (ω, d) its restriction when considering Γ_{K}/Γ_{F} instead of H. Obviously, res_{Γ_{K}/Γ_{F}}^{H} (ω, d) is a factor set of Γ_{K}/Γ_{F} in D_{0}.

(c) Let i : KUM(K_{0}/F_{0}) → D_{0}^{∗} be the inclusion map. For a cocycle h ∈ Z^{2}(Γ_{K}/Γ_{F} , KUM(K_{0}/F_{0})) we denote by i_{∗}h the map : Γ_{K}/Γ_{F} × Γ_{K}/Γ_{F} → D_{0}^{∗}, (γ̄, γ̄′) ↦ i ∘ h(γ̄, γ̄′).

Theorem 2. 4 Let F be a graded field, D a graded central division algebra over F, (ω, d) the factor set of Γ_{D}/Γ_{F} in D_{0} seen in Lemma 1.7, K a Kummer graded subfield of D and α_{K} the cocycle of Z^{2}(Γ_{K}/Γ_{F} , kum(K_{0}/F_{0}))_{sym} defined in (2.1), then there exists a cocycle d′ ∈ Z^{2}(Γ_{K}/Γ_{F} , KUM(K_{0}/F_{0}))_{sym} (for the trivial action of Γ_{K}/Γ_{F} on KUM(K_{0}/F_{0})) and a map ω′ : Γ_{K}/Γ_{F} → Aut(D_{0}) which satisfies ω′ ; γ̄(a) = a for all a ∈ K_{0} and γ̄ ∈ Γ_{K}/Γ_{F}, such that :

1. (ω′, i_{∗}d′) is a factor set of Γ_{K}/Γ_{F} in D_{0} cohomologous to res_{Γ_{K}/Γ_{F}}^{Γ_{D}/Γ_{F}} (ω, d), and

2. e_{∗}([d′]) = [α_{K}].

Proof. Let H = Γ_{D}/Γ_{F} and write D = ⊕_{γ̄∈H}D_{0}F xγ̄, where xγ̄a = ωγ̄(a)xγ̄ and xγ̄x_{}_{γ̄′} = d(γ̄, γ̄′)h(γ̄, γ̄′)x_{γ̄+γ̄′} (where h is the cocycle of Z^{2}(Γ_{D}/Γ_{F} , F^{∗})_{sym} seen in Lemma 1.7). For any γ ∈ Γ_{K}, let yγ̄ ∈ KUM(K/F) such that gr(yγ̄) + Γ_{F} = γ̄ (= γ + Γ_{F}) and write yγ̄ = aγ̄xγ̄, where aγ̄ ∈ (D_{0}F)^{∗}. Let bγ̄ ∈ D_{0}^{∗} and cγ̄ ∈ F^{∗} such that aγ̄ = bγ̄cγ̄, then we have :

yγ̄y_{}_{γ̄′} = aγ̄ωγ̄(a_{}_{γ̄′})d(γ̄, γ̄′)a_{γ̄+γ̄′}^{−1}h(γ̄, γ̄′)y_{γ̄+γ̄′}
= bγ̄ωγ̄(b_{}_{γ̄′})d(γ̄, γ̄′)b_{γ̄+γ̄′}^{−1}cγ̄c_{}_{γ̄′}c_{γ̄+γ̄′}^{−1}h(γ̄, γ̄′)y_{γ̄+γ̄′}
= d′(γ̄, γ̄′)h′(γ̄, γ̄′)y_{γ̄+γ̄′}

where d′(γ̄, γ̄′) = bγ̄ωγ̄(b_{}_{γ̄′})d(γ̄, γ̄′)b_{γ̄+γ̄′}^{−1} and h′(γ̄, γ̄′) = cγ̄c̄_{}_{γ̄′}c_{γ̄+γ̄′}^{−1}h(γ̄, γ̄′). Since yγ̄, y_{}_{γ̄′} and y_{γ̄+γ̄′} are in KUM(K/F) and h′(γ̄, γ̄′) ∈ F^{∗}, then d′(γ̄, γ̄′) ∈ KUM(K/F) ∩ D_{0} (= KUM(K_{0}/F_{0})). One can easily check that d′ ∈ Z^{2}(Γ_{K}/Γ_{F} , KUM(K_{0}/F_{0}))_{sym} (this follows from the equality (yγ̄y_{}_{γ̄′})y_{γ̄”} = yγ̄(y_{}_{γ̄′}y_{γ̄”}), the fact that h′ ∼ res_{Γ_{K}/Γ_{F}}^{H} (h) is a symmetric 2-cocycle and the fact that yγ̄ are pairwise commuting for γ̄ ∈ Γ_{K}/Γ_{F}). Now, let ω′ : Γ_{K}/Γ_{F} → Aut(D_{0}) be the map defined by ω′ ; γ̄ = Inn(bγ̄)ωγ̄ (i.e., ω′ ; γ̄(a) = bγ̄ωγ̄(a)b−1 ; γ̄ for all a ∈ D_{0} and γ̄ ∈ Γ_{K}/Γ_{F}). Then, for any a ∈ K_{0} and any γ̄ ∈ Γ_{K}/Γ_{F},

we have ω′ ; γ̄(a) = bγ̄xγ̄ax−1 ; γ̄b−1 ; γ̄ = aγ̄xγ̄ax−1 ; γ̄a−1 ; γ̄ = yγ̄ay−1 ; γ̄ = a. One can easily see that

(ω′, i_{∗}d′) is a factor set of Γ_{K}/Γ_{F} in D_{0} cohomologous to res_{Γ_{K}/Γ_{F}}^{H} (ω, d). Moreover, the equality yγ̄y_{}_{γ̄′} = d′(γ̄, γ̄′)h′(γ̄, γ̄′)y_{γ̄+γ̄′} yields, by considering classes modulo F^{∗} in kum(K/F), ȳγ̄ȳ_{}_{γ̄′} = e(d′(γ̄, γ̄′))ȳ_{γ̄+γ̄′}, where e : KUM(K_{0}/F_{0}) → kum(K_{0}/F_{0}) is the canonical surjective homomorphism (we identify here kum(K_{0}/F_{0}) with its canonical image in kum(K/F)). Hence, e_{∗}([d′]) = [α_{K}].

(2.5) Let F be a graded field, D a graded division algebra over F, A a finite abelian subgoup of D^{∗}/F^{∗} with exponent m, and for any a ∈ A, let d_{a} be a representative of a in D^{∗}. Assume that F_{0} contains a primitive m^{th} root of unity and let F (A) = F [d_{a} | a ∈ A] be the subring of D generated by F and the elements d_{a} (a ∈ A). If d_{a} are pairwise commuting, then as in the ungraded case F (A) is a Kummer graded field extension of F with kum(F (A)) = A (it suffices to see that F (A) is a graded field and that F rac(F (A)) = F rac(F)(A) when A is identified with its canonical image in Cq(D)^{∗}/F rac(F)^{∗}).

Conversely to Theorem 2.4, we have the following Theorem.

Theorem 2. 6 Let F be a graded field, D a graded central division algebra over F and (ω, d) the factor set of Γ_{D}/Γ_{F} in D_{0} seen in Lemma 1.7. Assume F_{0} contains enough roots of unity and that there are :

1. a field extension M of F_{0} in D_{0}, and a subgroup R of Γ_{D}/Γ_{F} acting trivially on M,

2. a cocycle d′ ∈ Z^{2}(R, KUM(M/F_{0}))_{sym} and a map ω′ : R → Aut(D_{0}) such that (ω′, i_{∗}d′) is a factor set of R in D_{0} cohomologous to res_{R}^{Γ_{D}/Γ_{F}} (ω, d) and such that ω′ ; γ̄(a) = a for all a ∈ M and γ̄ ∈ R. Then, there exists a Kummer graded subfield K of D such that :

1. K_{0} = M, Γ_{K}/Γ_{F} = R and

2. e_{∗}([d′]) = [α_{K}].

Proof. Let’s denote by H the quotient group Γ_{D}/Γ_{F} and write D = ⊕_{γ̄∈H}D_{0}F xγ̄, where xγ̄a = ωγ̄(a)xγ̄ and xγ̄x_{}_{γ̄′} = d(γ̄, γ̄′)h(γ̄, γ̄′)x_{γ̄+γ̄′} (h being the cocycle of Z^{2}(H, F^{∗})_{sym} seen in Lemma 1.7). The fact that (ω′, i_{∗}d′) is cohomologous to res_{R}^{H}(ω, d) means that there is a family (bγ̄)_{γ̄∈R} of elements of D_{0}^{∗} such that for all a ∈ D_{0}

and γ̄, γ̄′ ∈ R, we have ω′ ; γ̄(a) = bγ̄ωγ̄(a)b−1 ; γ̄ and d′(γ̄, γ̄′) = bγ̄ωγ̄(b_{}_{γ̄′})d(γ̄, γ̄′)b_{γ̄+γ̄′}^{−1}.

Let yγ̄ = bγ̄xγ̄ for all γ̄ ∈ R. Then, we have yγ̄y_{}_{γ̄′} = d′(γ̄, γ̄′)h(γ̄, γ̄′)y_{γ̄+γ̄′}. Let K = ⊕_{γ̄∈R}MF yγ̄(⊆ D). Since d′ and h are symmetric, then yγ̄ are pairwise commuting. Moreover, by hypotheses ω′ ; γ̄(a) = a for all a ∈ M and γ̄ ∈ R, so K is a commutative graded subring (hence a graded subfield) of D. Let A be the subgroup of D^{∗}/F^{∗} generated by kum(M/F_{0}) and the set {ȳγ̄}_{γ̄∈R}. One can easily see that up to a graded isomorphism we have K = F (A). Therefore, K is a Kummer graded field extension of F with kum(K/F) = A. Considering classes in kum(K/F), we have ȳγ̄ȳ_{}_{γ̄′} = e(d′(γ̄, γ̄′))ȳ_{γ̄+γ̄′}, where e : KUM(M/F_{0}) → kum(M/F_{0}) is the canonical surjective homomorphism (we identify here kum(M/F_{0}) with its canonical image in kum(K/F)), so kum(K/F) is the extension of kum(M/F_{0}) by R with cocycle e_{∗}([d′]).

(2.7) Let F be a graded field, D a semiramified graded division algebra over F and G = Gal(D_{0}/F_{0}). We know that Γ_{D}/Γ_{F} ≅ G. Therefore, any subgroup of Γ_{D}/Γ_{F} can be identified to a subgoup of G. Let’s consider the following diagram :

H^{2}(Γ_{K}/Γ_{F} , KUM(K_{0}/F_{0}))_{sym} →^{i_{∗}} H^{2}(Γ_{K}/Γ_{F} , D_{0}^{∗})
e_{∗} ↓  ↑ res_{Γ_{K}/Γ_{F}}^{G}
H^{2}(Γ_{K}/Γ_{F} , kum(K_{0}/F_{0}))_{sym}  H^{2}(G, D_{0}^{∗})

where i_{∗} is the homomorphism of cohomology groups induced by the inclusion map KUM(K_{0}/F_{0}) →^{i} D_{0}^{∗}, e_{∗} is the homomorphism of cohomology groups induced by the canonical surjective homomorphism e : KUM(K_{0}/F_{0}) → kum(K_{0}/F_{0}), and res_{Γ_{K}/Γ_{F}}^{G} is the restriction map. As a consequence of Theorem 2.4, we have the following Corollary :

Corollary 2. 8 Let F be a graded field, D a semiramified graded division algebra over F, G = Gal(D_{0}/F_{0}), d the cocycle of Z^{2}(G, D_{0}^{∗}) seen in Remark 1.8, K a Kummer graded subfield of D and α_{K} the cocycle of Z^{2}(Γ_{K}/Γ_{F} , kum(K_{0}/F_{0}))_{sym} defined in (2.1), then there exists a cocycle d′ ∈ Z^{2}(Γ_{K}/Γ_{F} , KUM(K_{0}/F_{0}))_{sym} such that :

(1) i_{∗}([d′]) = res_{Γ_{K}/Γ_{F}}^{G} ([d]), and

(2) e_{∗}([d′]) = [α_{K}].

Also, as a consequence of Theorem 2.6, we have the following Corollary.

Corollary 2. 9 Let F be a graded field, D a semiramified graded division algebra over F and d ∈ Z^{2}(G, D_{0}^{∗}) the cocycles seen in Remark 1.8. Assume F_{0} contains enough roots of unity and suppose there exist : a subfield M of D_{0} containing F_{0}, a subgroup R of Γ_{D}/Γ_{F} acting trivially on M, and a cocycle d′ ∈ Z^{2}(G, KUM(M/F_{0}))_{sym} such that i_{∗}([d′]) = res_{R}^{G}([d]). Then, there exists a Kummer graded subfield K of D such that :

(1) M = K_{0}, R = Γ_{K}/Γ_{F}, and

(2) [α_{K}] = e_{∗}([d′]).

(2.10) Now let E be a Henselian valued field and D a tame central division algebra over E such that char(Ē) does not divide deg(D). Since GD is a graded central division algebra over GE, then we can define a graded factor set (ω, d) corresponding to GD as made in Lemma 1.7. If K is a Kummer subfield of D, then by [HW(1), Theorem 5.2] GK is a Kummer graded subfield of GD. So, we can consider the symmetric cocycle α_{GK} of (2.1) corresponding to GK. For simplicity, we denote α_{GK} just by α_{K}. As a direct consequence of Theorem 2.4, we have the following Corollary

Corollary 2. 11 Let E be a Henselian valued field and D a tame central division algebra over E such that char(Ē) does not divide deg(D). Using the notations of (2.10), if K is a Kummer subfield of D, then there is a cocycle d′ ∈ Z^{2}(Γ_{K}/Γ_{E}, KUM(K̄/Ē))_{sym} (for the trivial action of Γ_{K}/Γ_{E} on KUM(K̄/Ē)) and a map ω′ : Γ_{K}/Γ_{E} → Aut(D̄) which satisfies ω′ ; γ̄(a) = a for all a ∈ K̄ and γ̄ ∈ Γ_{K}/Γ_{E}, such that :

1. (ω′, i_{∗}d′) is a factor set of Γ_{K}/Γ_{E} in D̄ cohomologous to res_{Γ_{K}/Γ_{E}}^{Γ_{D}/Γ_{E}}(ω, d), and

2. e_{∗}([d′]) = [α_{K}].

Also, as a consequence of Theorem 2.6, we have the following Corollary :

Corollary 2. 12 Let E be a Henselian valued field and D a tame central division algebra over E such that char(Ē) does not divide deg(D). Assume that Ē contains enough roots of unity and that (with the notations of (2.10)), there are :

1. a field extension M of Ē in D̄, and a subgroup R of Γ_{D}/Γ_{E} acting trivially on M,

2. a cocycle d′ ∈ Z^{2}(R, KUM(M/Ē))_{sym} and a map ω′ : R → Aut(D̄) such that (ω′, i_{∗}d) is a factor set of R in D̄ cohomologous to res_{R}^{ΓD/ΓE}(ω, d) and such that ω′ ; γ̄(a) = a for all a ∈ M and γ̄ ∈ R. Then, there exists a Kummer subfield K of D such that :

1. K̄ = M, Γ_{K}/Γ_{E} = R and

2. e_{∗}([d′]) = [α_{K}].

Remark 2.13 (1) In the last two corollaries, we can use the group isomorphism kum(K/E) ≅ kum(GK/GE) and replace the exact sequence of trivial Γ_{K}/Γ_{E}- modules α_{GK} by another exact sequence of trivial Γ_{K}/Γ_{E}-modules

1 → kum(K̄/Ē) →^{ϕ} kum(K/E) →^{ψ} Γ_{K}/Γ_{E} → 0

then use it to have necessary and sufficient condition for D to have Kummer subfields.

(2) We have also analogous results to Corollary 2.8 and Corollary 2.9 for tame semiramified division algebras over Henselian valued fields.

(3) We can drop the assumption that E is Henselian in many results of this paper. Indeed, let D be a valued central division algebra over a field E, HE be the Henselization of D with respect to the restriction of the valuation of D and HD = D ⊗_{E} HE. Then, one can easily see that GD = G(HD) and GE = G(HE).

Theorem 2. 14 Let F be a graded field, D a semiramified graded division algebra over F and d the cocycle seen in Remark 1.8. If F_{0} contains a primitive deg(D)^{th} root of unity, then the following statements are equivalent :

(1) D is cyclic,

(2) There is a field extension M of F_{0} in D_{0} such that :

(i) the extensions M/F_{0} and D_{0}/M are cyclic, and

(ii) (D_{0}/F_{0}, G, d) ⊗_{F_{0}} M ∼ (D_{0}/M, σ, u) for some generator σ of Gal(D_{0}/M) and some u ∈ M^{∗} such that uF_{0}^{∗} generates kum(M/F_{0}).

Proof. This can be proved in the same way as [T86, Theorem 3.1].

Theorem 2. 15 Let F be a graded field, D a semiramified graded division algebra over F and d the cocycle seen in Remark 1.8. Suppose now that deg(D) is a power of a prime p and that F_{0} contains a primitive p^{th} root of unity. Then, the following statements are equivalent

(1) D is an elementary abelian graded crossed product,

(2) there is a field extension M of F_{0} in D_{0} such that M/F_{0} and D_{0}/M are elementary abelian, and (D_{0}/F_{0}, G, d) represents in Br(D_{0}/F_{0})/Dec(D_{0}/F_{0}) an element of the image of the canonical group homomorphism Br(M/F_{0})/Dec(M/F_{0}) → Br(D_{0}/F_{0})/Dec(D_{0}/F_{0}),

(3) exp(G) = p or p^{2} and (D_{0}/F_{0}, G, d) represents in Br(D_{0}/F_{0})/Dec(D_{0}/F_{0}) an element of the image of the canonical group homomorphism Br(L/F_{0})/Dec(L/F_{0}) → Br(D_{0}/F_{0})/Dec(D_{0}/F_{0}), where L = F ix_{G^{p}}(D_{0}) (G^{p} being the subgoup of G consisting in p-powers of elements of G) (this last condition is void if exp(G) = p since in this case L = K.)

Proof. This can be proved in the same way as [T86, Theorem 4.1].

Proposition 2. 16 Let E be a Henselian valued field, D a division algebra over E such that char(Ē) does not divide deg(D) and H a finite group. Then, D has a tame Galois subfield with Galois group isomorphic to H if and only if GD has a Galois graded subfield of Galois group isomorphic to H. Therefore, D is cyclic [resp., an elementary abelian crossed product] if and only if GD is cyclic [resp., an elementary abelian graded crossed product].

Proof. Assume that D has a Galois subfield of Galois group isomorphic to H, then by [HW(1), Theorem 5.2] GK is a Galois graded subfield of GD with Galois group isomorphic to H. Conversely, assume that GD has a Galois graded subfield L with Galois group isomorphic to H. Then, again by [HW(1), Theorem 5.2] there is a tame field extension M of E such that GM ≅ L and Gal(M/E) ≅ H. By [HW(2)99, Theorem 5.9] M is isomorphic to a subfield of D.

Remark. We recall that if E is a Henselian valued field and D is an inertially split division algebra over E with D̄ commutative, then D is a tame semiramified division algebra over E (see [M07, Proposition 2.6]). The reader can then see that similar results to Theorem 2.14, Theorem 2.15 in the case of tame semiramified division algebras over a Henselian valued field were proved in [MorSe95]. Using Theorem 2.14, Theorem 2.15, we get the next two Corollaries of [MorSe95]. In the next section, we will prove these two corollaries without assuming that Ē contains primitive roots of unity.

Corollary 2. 17 [MorSe95, Corollary 5.5] Let E be a Henselian valued field and D a tame semiramified division algebra of prime power degree over E. Suppose that char(Ē) does not divide deg(D) and Ē contains a primitive deg(D)^{th} root of unity and that rk(Γ_{D}/Γ_{E}) ≥ 3, then D is non-cyclic.

Proof. We have rk(Gal(GD_{0}/GE_{0})) = rk(Gal(D̄/Ē)) = rk(Γ_{D}/Γ_{E}) ≥ 3. So by Theorem 2.14(2(i)) GD is non-cyclic. Hence, by Proposition 2.16, D is non-cyclic.

Corollary 2. 18 [MorSe95, Corollary 5.7] Let E be a Henselian valued field and D a tame semiramified division algebra of prime power degree p^{n} over E (p being a prime integer and n ∈ IN^{∗}). Suppose that Ē contains a primitive p^{th} root of unity and that p^{3} divides exp(Γ_{D}/Γ_{E}), then D has no elementary abelian maximal subfield.

Proof. This follows by Theorem 2.15 and Proposition 2.16.

3 Non-cyclic and non-elementary abelian crossed product tame semiramified division algebras

Let E be a Henselian valued field and D a tame semiramified division algebra of prime power degree p^{n} over a Henselian valued field E such that char(Ē) ≠ p. In this section, we aim to show that if rk(Γ_{D}/Γ_{E}) ≥ 3, then D is non-cyclic [Proposition 3.1], and that if p^{3} divides exp(Γ_{D}/Γ_{F}), then D has no elementary abelian maximal subfield [Proposition 3.2].

Proposition 3. 1 Let E be a Henselian valued field and D a semiramified division algebra of degree n over E. Assume char(Ē) does not divide n and suppose K is a cyclic maximal subfield of D. Then, Γ_{K}/Γ_{E} and Γ_{D}/Γ_{K} are cyclic. So, Γ_{D}/Γ_{E} is generated by two elements. In particular, if n is a prime power and rk(Γ_{D}/Γ_{E}) ≥ 3, then D is non-cyclic.

Proof. Let M be the inertial lift of K̄ over E in K (see [JW90, Theorem 2.8 and Theorem 2.9]). Since K is cyclic and totally ramified over M, then Γ_{K}/Γ_{E}(= Γ_{K}/Γ_{M}) is cyclic. Furthermore, we have Γ_{D}/Γ_{K} ≅ (Γ_{D}/Γ_{E})/(Γ_{K}/Γ_{E}) ≅ Gal(D̄/Ē)/Gal(D̄/K̄) ≅ Gal(K̄/Ē) ≅ Gal(M/E) (for the second equivalence, see that K is a totally ramified maximal subfield of the semiramified division algebra C_{D}^{M}). So, Γ_{D}/Γ_{K} is cyclic. Let γ_{1} + Γ_{E} be a generator of Γ_{K}/Γ_{E} and γ_{2} + Γ_{K} a generator of Γ_{D}/Γ_{K}, then for any α ∈ Γ_{D}/Γ_{E}, there are positive integers n_{1} and n_{2} such that α = n_{1}γ_{1} + n_{2}γ_{2} + Γ_{E}. If n is a prime power, then rk(Γ_{D}/Γ_{E}) ≤ 2.

Proposition 3. 2 Let E be a Henselian valued field and D a tame semiramified division algebra of prime power degree p^{n} over E (p being a prime integer and n ∈ IN^{∗}). If char(Ē) ≠ p and p^{3} divides exp(Gal(D̄/Ē)), then D has no elementary abelian maximal subfield.

Proof. Suppose that K is an elementary abelian maximal subfield of D, then K̄/Ē is elementary abelian. Therefore, for any σ ∈ Gal(D̄/Ē), σ^{p} ∈ Gal(D̄/K̄). Let M be the inertial lift of K̄ over E in K. Then, K is a Galois totally ramified field extension of M and Gal(K/M) ≅ Γ_{K}/Γ_{M}. Moreover, since C_{D}^{M} is tame semiramified, then Gal(D̄/K̄) = Gal(D̄/M̄) ≅ Γ_{K}/Γ_{M}(≅ Gal(K/M)). Hence, σ^{p^{2}} = idD̄. A contradiction.

Remark 3.3 (1) We recall that we saw in [M07, Proposition 4.6] that if E is a Henselian valued field and D is a nondegenerate tame semiramified division algebra of prime power degree over E, then D has an elementary abelian maximal subfield if and only if Γ_{D}/Γ_{F} is elementary abelian.

(2) As showed in [T86] with Malcev-Neumann division algebras, one can use Proposition 3.1 and Proposition 3.2 to prove the following result : Let m and n be integers which have the same prime factors and such that m divides n, and let k be an infinite field. If there is a prime p ≠ char(k) such that p^{2} divides m and p^{3} divides n, then Saltman’s universal division algebras of exponent m and degree n over k are not crossed products.

References

[BM00] M. Boulagouaz and K. Mounirh, Generic abelian crossed products and graded division algebras, pp. 33-47 in algebra and number theory (Fès, 1997), Lecture Notes in Pure and Appl. Math. Vol. 208, Dekker, New York, 2000.

[JW90] B. Jacob and A. R. Wadsworth, Division algebras over Henselian Fields, J. Algebra, 128 (1990), pp. 126-179.

[HW(1)99] Y.-S. Hwang and A. R. Wadsworth, Algebraic extensions of graded and valued fields, Comm. Algebra, 27 (1999), pp. 821-840.

[HW(2)99] Y.-S. Hwang and A. R. Wadsworth, Correspondences between valued division algebras and graded division algebras, J. Algebra, 220 (1999), pp. 73-114.

[MorSe95] P. Morandi and B.A. Sethuraman, Kummer subfields of tame division algebras, J. Algebra, 172 (1995), pp. 554-583.

[M05] K. Mounirh, Nicely semiramified division algebras over Henselian fields, International Journal of Mathematics and Mathematical Sciences (2005), 571-577.

[M07] K. Mounirh, Nondegenerate semiramified valued and graded division algebras, May 2007, Preprint (LAG).

[NO82] C. Natasescu and F. Van Oystaeyen, Graded ring Theory, North-Holland, Library of Math., Vol. 82, 1982.

[P82] R-S. Pierce, Associative algebra, Spring-Verlag (1982).

[TA85] JP Tignol and SA Amitsur, Kummer subfields of Malcev-Neumann division algebras, Isr. Journal of Math. Vol. 50 (1985), pp. 114-144.

[T86] JP Tingol, Cyclic and elementary abelian subfields of Malcev-Neumann division algebras, Journal of Pure and appl. algebra, Vol 42 (1986), pp. 199-220.

[T87] JP Tignol, Generalized crossed products, in ”Seminaires Mathématiques” (nouvelle série), N106, Université Catholique de Louvain-La-Neuve, Belgium, 1987.