CENTRAL SIMPLE ALGEBRAS OF PRIME

EXPONENT AND DIVIDED POWER OPERATIONS

A.S. Sivatski

Departamento de Matemática Universidade Federal do Ceará Fortaleza, Brasil

Abstract. Let p be a prime, F a field of characteristic different from p. We prove triviality of the divided power operations on central simple cyclic algebras of exponent p.

Introduction

Let F be a field, Br(F) the Brauer group of F, α ∈ Br(F). A natural question arises whether α can be represented by a cyclic algebra. For instance, if ind α = 2, then the answer is always positive, since, as is well-known, in this case α is represented by a quaternion algebra. If char F ≠ 2, exp α = 2, ind α = 4, and \sqrt{−1} ∈ F, then α = (a, b) + (c, d) is represented by a cyclic algebra of degree 4 if and only if the Pfister form ⟨⟨a, b, c, d⟩⟩ is hyperbolic, or, in other words, {a, b, c, d} = 0 ∈ K_{4}(F)/2K_{4}(F) ([RST], Th.3). However, the question of representation of α by a cyclic algebra of higher degree was not investigated in [RST]. In the present paper we give necessary conditions for an element α ∈ Br(F) of arbitrary prime exponent p to be cyclic, provided all p-primary roots of unity are contained in F.

A few words about the notation. If n is a positive integer, a, b ∈ F^{∗}, char F |̸n, and ξ_{n} is a fixed primitive root of unity lying in F, then by (a, b)_{n} we denote the symbol algebra of degree n with the generators i and j, and the relations i^{n} = a, j^{n} = b, and ij = ξ_{n}ji. (If n = 2, then we write just (a, b) instead of (a, b)_{2}). Slightly abusing notation, by the same symbol we also denote the corresponding element in the Brauer group. If n = km and ξ_{n} ∈ F, then k(a, b)_{n} = (a, b)_{m} ∈ Br(F), where the roots of unity ξ_{n} and ξ_{m} are related by the equality ξ_{n}^{k} = ξ_{m}.

Given m ≥ 0, by K_{m}(F) we denote the mth Milnor K-group of the field F. Let n be a fixed positive integer and −1 ∈ F^{∗}^{n}. By γ_{k} : K_{2}(F)/n → K_{2k}(F)/n we denote the divided power operation of degree k on K_{2}(F)/n well defined by the following rule:

γ_{k}(∑_{i} α_{i}) = ∑_{i_{1}<i_{2}<···<i_{k}} α_{i_{1}}α_{i_{2}} … α_{i_{k}},

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16K50, 19D45.

Key words and phrases. Cyclic algebras, Milnor K-groups, Divided power operations, Norm residue isomorphism.

where all α_{j} are symbols ([K], [Vi]). Obviously, the divided power operations commute with field extension homomorphisms.

Given n ≥ 2, by _{n}Br(F) we denote the n-torsion of Br(F). If ξ_{n} ∈ F, then the norm residue map K_{2}(F)/n → _{n}Br(F), taking {a, b} to (a, b)_{n} is an isomorphism([MS], Th. 11.5), which allows us to define in the obvious way the divided power operations γ_{i} : _{n}Br(F) → K_{2i}(F)/n (provided that −1 ∈ F^{∗}^{n}). These operations depend on the choice of ξ_{n}, and change of ξ_{n} multiplies γ_{i} by m^{i}, where m is a positive integer coprime to n.

We outline briefly the main results from Milnor’s K-theory used in the paper. All of them except the norm residue isomorphism theorem (see below) can be found in [GS], Ch.7. One of the main tools is the exact sequence

0 → K_{m}(F)/n ⟶^{res} K_{m}(F (t))/n ⟶^{⊕∂_{f}} ∐_{f} K_{m−1}(F_{f})/n → 0, (∗)

where f runs over all monic irreducible polynomials over F, and F_{f} = F [t]/(f). The residue homomorphism ∂_{f} here is well defined by the following rule: if v_{f}(g_{i}) = 0 for all 2 ≤ i ≤ m, then ∂_{f}\{g_{1}, g_{2}, … g_{m}\} = v_{f}(g_{1})\{\overline{g_{2}}, …, \overline{g_{m}}\}, where v_{f} is the valuation of F (t) associated with f, and \overline{g_{i}} is the image of g_{i} in F_{f}. This sequence splits by ”the leading coefficient map” K_{m}(F (t))/n ⟶^{l} K_{m}(F)/n, where l({f_{1}, …, f_{m}}) = {l(f_{1}), …, l(f_{m})}, f_{i} ∈ F [t] and l(f_{i}) is the leading coefficient of f_{i}. In particular, the sequence

0 → _{n}Br(F) ⟶^{res} _{n}Br F (t) ⟶^{⊕∂_{f}} ∐_{f} F_{f}^{∗}/F_{f}^{∗}^{n} → 1,

is split exact, provided ξ_{n} ∈ F and char F |̸n.

To make some computations in the second part of the paper shorter we apply the Weyl reciprocity law for K_{m}(F (t))/2, namely the exact sequence

0 → k_{m}(F) ⟶^{res} k_{m}(F (t)) ⟶^{⊕∂_{f}} ⊕k_{m−1}(F_{f}) ⟶^{⊕N_{F_{f} /F}} k_{m−1}(F) → 0,

where k_{m} = K_{m}/2, f runs over all monic irreducible polynomials over F together with the infinity point, and N_{F_{f} /F} are norm maps. Also we need the projection formula N_{L/F} (res_{L/F} α · β) = α · N_{L/F} (β) for a finite extension L/F, α ∈ K_{∗}(F)/p, β ∈ K_{∗}(L)/p. In particular, if L/F is a quadratic extension, and α ∈ K_{∗}(F), then setting β = \overline{1} ∈ K_{0}(L)/2, we get N_{L/F} (res_{L/F} α) = 0.

The only ”nonelementary” and indeed, very deep result applied in the paper is the exact sequence

K_{∗}(F)/p ⟶^{mult.by\{a\}} K_{∗+1}(F)/p ⟶^{res} K_{∗+1}(F (\sqrt[p]{a}))/p,

for a cyclic extension F (\sqrt[p]{a})/F provided μ_{p} ∈ F^{∗}. Exactness of this sequence can be inferred from the norm residue isomorphism K_{∗}(F)/p ≃ H^{∗}(F, μ_{p}), proven by Rost and Voevodsky, ([V], [W]).

1. Cyclic algebras of prime exponent

The first part of the paper deals with cyclic algebras of prime exponent. We show that all divided power operations are trivial on them, provided the ground field contains all p-primary roots of unity. More precisely we have the following

Theorem 1.1. Let p be a prime, F a field, char F ≠ p, m positive integer, ξ_{p^{m}} ∈ F, α ∈ _{p} Br(F). Suppose a ∈ F^{∗} is such that F ( \sqrt[p^{m}]{a}) is a splitting field for α, i.e. α_{F ( \sqrt[p^{m}]{a})} = 0. Then γ_{i}(α) = 0 for any i ≥ 2.

Proof. We will induct on m, the case m = 1 being trivial, since α corresponds then to a symbol in K_{2}(F)/p. By the hypothesis α = (a, b)_{p^{m}} for some b ∈ F^{∗} ([P], §§ 13.3, 15.1 ). Since exp α = p, we get (a, b)_{p^{m−1}} = 0, which means that b = N_{F ( \sqrt[p^{m−1}]{a})/F} (c) for some c ∈ F ( \sqrt[p^{m−1}]{a}) ([P], § 15.1). Consider first the generic case, where x_{0}, … x_{p^{m−1}−1}, a are indeterminates over a prime field k, F =

k(x_{0}, …, x_{p^{m−1}−1}, a) and b = N_{F (  \sqrt[p^{m−1}]{a})/F} ( ∑_{j=0}^{p^{m−1}−1} x_{j}  \sqrt[p^{m−1}]{a^{j}}) ∈ k[x_{0}, …, x_{p^{m−1}−1}, a]

is the norm polynomial of the generic element of F ( \sqrt[p^{m−1}]{a}). Set K = k(x_{0}, …, x_{p^{m−1}−1}), so that F = K(a). The commutative diagram

p Br K(a) \underrightarrow{∐ ∂_{f}} ∐_{f} K_{f}^{∗}/K_{f}^{∗}^{p}
id↓  ^{p^{m−1}}↓
p^{m}Br K(a) \underrightarrow{∐ ∂_{f}} ∐_{f} K_{f}^{∗}/K_{f}^{∗}^{p^{m}}

shows that α has nonzero residues at the same polynomials independently of whether it is considered as an element of _{p} Br K(a) or of _{p^{m}}Br K(a). Hence, α ∈ _{p} Br K(a) can have nonzeros residues only at the monic polynomials a and b′ = l(b)^{−1}b, where l(b) is the leading coefficient of b. Moreover, the similar argument shows that the image of α under ”the leading coefficient” map l : _{p} Br K(a) → _{p} Br K is zero.

In view of the exact sequence (∗) the assertion that γ_{i}(α) = 0 is equivalent to that ∂_{f}(γ_{i}(α)) = 0 for any monic polynomial f ∈ K[a], and l(γ_{i}(a)) = 0. This is just what we are going to prove.

Lemma 1.2. Let i ≥ 2. Then

1) ∂_{f}(γ_{i}(α)) = 0 for any monic irreducible polynomial f ∈ K[a] distinct from a and b′.

2) l(γ_{i}(α)) = 0.

Proof. 1) The element α ∈ _{p} Br(K(a)) can be written as α = (f, g)_{p} + ∑_{j}(f_{j}, g_{j})_{p}

where g, f_{j}, g_{j} ∈ K[a] are polynomials not divisible by f. Since ∂_{f}(α) = 0, we have g ∈ F_{f}^{∗}^{p}, which makes the assertion obvious.

2) This follows from the facts that l(a) = 1 and the commutative diagram

K_{2}(K(a))/p ⟶^{l} K_{2}(K)/p
γ_{i}↓  ^{γ_{i}}↓
K_{2i}(K(a))/p ⟶^{l} K_{2i}(K)/p

The lemma is proved. □

It remains to show that ∂_{a}(γ_{i}(α)) = 0 and ∂_{b′}(γ_{i}(α)) = 0. To do this represent α as α = (a, x)_{p}+α_{1}, where x = ∂_{a}(α) ∈ K, and ∂_{f}(α_{1}) = 0 for any monic polynomial f ≠ b′. Obviously, α_{1}_{F ( \sqrt[p^{m}]{a})} = α_{F ( \sqrt[p^{m}]{a})} = 0, hence γ_{i−1}(α_{1})_{F ( \sqrt[p^{m}]{a})} = 0. Since by the projection formula

\{a\}γ_{i−1}(α_{1}) = N_{F ( \sqrt[p^{m}]{a})/F} ( \sqrt[p^{m}]{a})γ_{i−1}(α_{1})_{F ( \sqrt[p^{m}]{a})} = N_{F ( \sqrt[p^{m}]{a})/F} (0) = 0,

we have

γ_{i}(α) = {a, x}γ_{i−1}(α_{1}) + γ_{i}(α_{1}) = γ_{i}(α_{1}).

Since ∂_{a}(α_{1}) = 0, it follows that ∂_{a}(γ_{i}(α)) = ∂_{a}(γ_{i}(α_{1})) = 0. Furthermore, by the induction hypothesis applied to the element α_{F ( \sqrt[p]{a})}, we have

γ_{i}(α)_{F ( \sqrt[p]{a})} = γ_{i}(α_{F ( \sqrt[p]{a})}) = 0.

By the result of Voevodsky mentioned in the introduction ([V], [W]), we conclude that γ_{i}(α) = {a}β for some β ∈ K_{2i−1}(F)/p.

Therefore,

∂_{b′}(γ_{i}(α)) = ∂_{b′}({a}β) = {a}∂_{b′}(β) = 0,

since a ∈ K_{b′}^{∗}^{p} in view of the equality (a, b)_{p} = 0.

Thus, we have proved the theorem in the generic case. In the general case con-

sider any element α̃ ∈ _{p} Br(F). We have α̃ = (ã, b̃)_{p^{m}}, where b̃ = N_{F (  \sqrt[p^{m−1}]{ã})/F} ( ∑_{j=0}^{p^{m−1}−1} x̃_{j}  \sqrt[p^{m−1}]{a^{j}}),

and ã, x̃_{j} ∈ F. Interpreting the groups _{p} Br and _{p^{m}}Br as K_{2}/p and K_{2}/p^{m} respectively, we obtain the following commutative diagrams

p Br(k(x_{0}, …, x_{p^{m−1}−1}, a)) ⟶^{s_{p}} _{p} Br(F)
id↓  ^{id}↓
p^{m}Br(k(x_{0}, …, x_{p^{m−1}−1}, a)) ⟶^{s_{p^{m}}} _{p^{m}}Br(F)

and

p Br(k(x_{0}, …, x_{p^{m−1}−1}, a)) ⟶^{s_{p}} _{p} Br(F)
γ_{i}↓  ^{γ_{i}}↓
K_{2i}(k(x_{0}, …, x_{p^{m−1}−1}, a))/p ⟶^{s_{p}} K_{2i}(F)/p

where s_{p} and s_{p^{m}} are specialization maps taking x_{j} to x̃_{j} and a to ã. Looking at the first diagram we see that α̃ = s_{p}(α), where α is the element considered in the generic case. Then the second diagram yields

γ_{i}(α̃) = γ_{i}(s_{p}(α)) = s_{p}(γ_{i}(α)) = s_{p}(0) = 0.

This finishes the proof of Theorem 1.1. □

Remark. We do not know how to avoid using the higher norm residue isomorphism to obtain equality γ_{i}(α) = {a}β, which is essential in the proof of Theorem 1.1. Notice also that this proof can not be generalized straightforwardly to the case of an arbitrary p-primary exponent.

We will call an element α ∈ Br(F) cyclic if α_{L} = 0 for some cyclic field extension L/F.

Corollary 1.3. Let F be a field, p a prime, char F ≠ p. Suppose all p-primary roots of unity are contained in F, and α ∈ _{p} Br F is cyclic. Then γ_{i}(α) = 0 for any i ≥ 2.

Proof. By the hypothesis α_{L} = 0 for some cyclic field extension L/F. Then α_{K} = 0 for some intermediate field K between F and L such that [K : F] = p^{m} and [L : K] is not divisible by p. Then K = F ( \sqrt[p^{m}]{a}) for some a ∈ F^{∗} and the result follows from Theorem 1.1. □

Corollary 1.4. Let char F ≠ 2, D a biquaternion division algebra over F, m a positive integer, ξ_{2^{m}} ∈ F, a ∈ F^{∗}. Suppose D_{F ( \sqrt[2^{m}]{a})} = 0 ∈ Br(F). Then D is a cyclic algebra.

Proof. By Theorem 1.1 we have γ_{2}(D) = 0, which, on the other hand, is equivalent to the assertion that the algebra D is cyclic ([RST], Th.3). □

The statement converse to Theorem 1.1 is not true, as the following counterexample shows.

Proposition 1.5. Let p be a prime, k a field, containing all p-primary roots of unity, char k ≠ p. Suppose that a, b, c ∈ k^{∗} are such that [k(\sqrt[p]{a}, \sqrt[p]{b}, \sqrt[p]{c}) : k] = p^{3}. Let further F = k((x))((y))((z)) be the iterated Laurent series field over k. Then the following holds:

1) The element D = (a, x)_{p} + (b, y)_{p} + (c, z)_{p} is not cyclic.

2) If, in addition, (a, b)_{p} = (a, c)_{p} = (b, c)_{p} = 0 (this condition holds, in particular, for any field k with cd_{p} k = 1; for instance, one can put k = ℂ(t), a = t, b = t + 1, c = t + 2), then γ_{i}(D) = 0 for any i ≥ 2.

Proof. 1) Assume the converse. The p-primary component of any cyclic element is cyclic as well. Furthermore, for any field L of characteristic not p there is a natural isomorphism of groups L^{∗}/L^{∗}^{p} ⊕ℤ/pℤ ≃ L((t))^{∗}/L((t))^{∗}^{p} given by the rule (\overline{l}, \overline{i}) → \overline{lt^{i}} (0 ≤ i ≤ p − 1). Therefore, we may assume that

D = (a^{p^{m−1}}, x)_{p^{m}} + (b^{p^{m−1}}, y)_{p^{m}} + (c^{p^{m−1}}, z)_{p^{m}} = (αx^{i_{1}}y^{j_{1}}z^{k_{1}}, βx^{i_{2}}y^{j_{2}}z^{k_{2}})_{p^{m}}

for some α, β ∈ k^{∗}. Taking the residue at z we get

(−1)^{k_{1}k_{2}}(αx^{i_{1}}y^{j_{1}})^{k_{2}}(βx^{i_{2}}y^{j_{2}})^{−k_{1}} = c^{p^{m−1}}f^{p^{m}}

for some f ∈ k((x))((y)). It follows that (−1)^{k_{1}k_{2}}α^{k_{2}}β^{−k_{1}} = c^{p^{m−1}}u^{p^{m}} for some u ∈ k^{∗}, which gives us, in particular, that \sqrt[p]{c} ∈ k( \sqrt[p^{m}]{α}, \sqrt[p^{m}]{β}). Moreover, (a^{p^{m−1}}, x)_{p^{m}} + (b^{p^{m−1}}, y)_{p^{m}} = (αx^{i_{1}}y^{j_{1}}, βx^{i_{2}}y^{j_{2}})_{p^{m}}. Applying the above argument again, we subsequently get \sqrt[p]{b} ∈ k( \sqrt[p^{m}]{α}, \sqrt[p^{m}]{β}) and \sqrt[p]{a} ∈ k( \sqrt[p^{m}]{α}, \sqrt[p^{m}]{β}), i.e. we obtain a field embedding k(\sqrt[p]{a}, \sqrt[p]{b}, \sqrt[p]{c}) ⊂ k( \sqrt[p^{m}]{α}, \sqrt[p^{m}]{β}). This is a contradiction, since the Galois group of the extension k( \sqrt[p^{m}]{α}, \sqrt[p^{m}]{β})/k is generated by two elements, but the Galois group of the extension k(\sqrt[p]{a}, \sqrt[p]{b}, \sqrt[p]{c})/k is not.

2) This part of the proposition is obvious in view of the equalities

γ_{2}(D) = {a, b, x, y} + {a, c, x, z} + {b, c, y, z} ∈ K_{4}(F)/p

and γ_{3}(D) = {a, b, c, x, y, z} ∈ K_{6}(F)/p. □

A natural question arises whether it is possible to generalize Theorem 1.1 to arbitrary primary exponent and corresponding divided power operations. We do not know the answer, but the following statement shows that at least this is the case for cyclic algebras of degree 2^{n} and exponent 2^{n−1}.

Proposition 1.6. Suppose an element D ∈ Br(F) is represented by a cyclic algebra of degree 2^{n} and exponent 2^{n−1}, and ξ_{2^{n}} ∈ F. Then γ_{i}(D) = 0 for any i ≥ 2, where γ_{i} : K_{2}(F)/2^{n−1} → K_{2i}(F)/2^{n−1} are the corresponding divided power operations.

Proof. By the hypothesis D = (a, x^{2} − ay^{2})_{2^{n}}. We have

\begin{matrix}D = (a, x^{2} − ay^{2})_{2^{n}} = (a, x^{2})_{2^{n}} + (a, 1 − ax^{−2}y^{2})_{2^{n}} = (a, x^{2})_{2^{n}} + (x^{2}y^{−2}, 1 − ax^{−2}y^{2})_{2^{n}}+ \\ (ax^{−2}y^{2}, 1 − ax^{−2}y^{2})_{2^{n}} = (a, x)_{2^{n−1}} + (xy^{−1}, 1 − ax^{−2}y^{2})_{2^{n−1}}. \end{matrix}

Therefore,

γ_{2}(D) = {a, x, xy^{−1}, 1 − ax^{−2}y^{2}} = {x^{2}y^{−2}, x, xy^{−1}, 1 − ax^{−2}y^{2}}+

{ax^{−2}y^{2}, x, xy^{−1}, 1 − ax^{−2}y^{2}} = {x^{2}y^{−2}, x, xy^{−1}, 1 − ax^{−2}y^{2}} =

2{xy^{−1}, x, xy^{−1}, 1 − ax^{−2}y^{2}} = 2{−1, x, xy^{−1}, 1 − ax^{−2}y^{2}} = 0 ∈ K_{4}(F)/2^{n−1}

since −1 ∈ F^{∗}^{2^{n−1}}, and, moreover, γ_{i}(D) = 0 if i ≥ 3, since D = (a, x)_{2^{n−1}} + (xy^{−1}, 1 − ax^{−2}y^{2})_{2^{n−1}}. The proposition is proved. □

2. Division algebras of index 8 and exponent 2

In the second part of the paper we apply the division power operations γ_{2} and γ_{3} to studying the structure of a central division algebra D of exponent 2 and index 8 over F. Theorem 1.1 shows that the condition γ_{2}(D) = γ_{3}(D) = 0 is necessary for existense of a specific (cyclic) maximal subfield of D. It turns out that the condition γ_{3}(D) = 0 is necessary for existence of a specific subfield of D of degree

4. The key point is a certain relationship between γ_{2}(D) and γ_{3}(D).

In the sequel we denote K_{∗}/2 by k_{∗}, and all symbols are supposed to be in k_{∗}. The isomorphism k_{2} ≃_{2} Br allows us to multiply elements from _{2} Br by elements from k_{∗}. We start with the following

Theorem 2.1. Let F be a field, char F ≠ 2, \sqrt{−1} ∈ F^{∗}. Suppose D ∈_{2} Br(F), ind D = 8, and a ∈ F^{∗} is such that ind D_{F (\sqrt{a})} = 4. Then

1) {a}γ_{2}(D) ∈ k_{5}(F) is a symbol.

2) There exists s ∈ F^{∗} such that γ_{3}(D) = {a, s}γ_{2}(D) ∈ k_{6}(F). In particular, γ_{3}(D) is a symbol.

Proof. 1) It follows from the proof of Theorem 6.2 in [R] that there exist b, c ∈ F^{∗} such that D_{F (\sqrt{a},\sqrt{b},\sqrt{c})} = 0. Hence

D_{F (\sqrt{a})} = (b, ω_{1}) + (c, ω_{2}),

where ω_{1}, ω_{2} ∈ F (\sqrt{a}) ([ELW]), and, consequently, since N_{F (\sqrt{a})/F} D_{F (\sqrt{a})} = 0, we get (b, N_{F (\sqrt{a})/F} ω_{1}) = (c, N_{F (\sqrt{a})/F} ω_{2}). Therefore,

N_{F (\sqrt{a})/F} \{b, c, ω_{1}\} = \{b, c, N_{F (\sqrt{a})/F} ω_{1}\} = \{c, c, N_{F (\sqrt{a})/F} ω_{2}\} = 0.

By the transfer principle ([EL], 2.2) and the isomorphism k_{3} ≃ I^{3}/I^{4} it can be easily seen that \{b, c, ω_{1}\} = \{b, c, r\}_{F (\sqrt{a})} for some r ∈ F^{∗}. Similarly, \{b, c, r, ω_{2}\} = \{b, c, r, t\}_{F (\sqrt{a})} for some t ∈ F^{∗}. Hence

γ_{2}(D)_{F (\sqrt{a})} = \{b, c, ω_{1}, ω_{2}\}_{F (\sqrt{a})} = \{b, c, r, t\}_{F (\sqrt{a})}.

It follows that γ_{2}(D) = {b, c, r, t}+{a}θ for some θ ∈ k_{3}(F). Therefore, {a}γ_{2}(D) = {a, b, c, r, t}, which proves part 1).

2) The proof of the second part of the theorem was proposed by the referee. Let B be the centralizer of F (\sqrt{a}) in D. The trace form q_{B}(x) = Trd(x^{2}) is a 4-fold Pfister form over F (\sqrt{a}) corresponding to γ_{2}(B) = γ_{2}(D_{F (\sqrt{a})}) under the isomorphism k_{4}(F (\sqrt{a})) ≃ I^{4}(F (\sqrt{a}))/I^{5}(F (\sqrt{a})). As was shown in the first part of Theorem 2.1, we have γ_{2}(D_{F (\sqrt{a})}) = \{b, c, r, t\} for some b, c, r, t ∈ F^{∗}. Hence T_{F (\sqrt{a})/F} (q_{B}) = ⟨2⟩⟨⟨a, b, c, r, t⟩⟩, where

T_{F (\sqrt{a})/F} : W (F (\sqrt{a})) → W (F)

is the transfer, corresponding to the trace for the extension F (\sqrt{a})/F. It was shown ([BM], proof of Th. 2.10) that the trace form q_{D}(x) = Trd(x^{2}) contains T_{F (\sqrt{a})/F} (q_{B}) as a subform, and that q_{D} is a multiple of a 6-fold Pfister form. It follows that q_{D} = ⟨2⟩⟨⟨a, b, c, r, t, s⟩⟩ for some s ∈ F^{∗}. By [BM], Th. 2.10 the form q_{D} corresponds to γ_{3}(D) under the isomorphism k_{6}(F) ≃ I^{6}(F)/I^{7}(F), hence γ_{3}(D) = {a, b, c, r, t, s}. Since by the first part of Theorem 2.1 {a}γ_{2}(D) = {a, b, c, r, t}, we conclude that γ_{3}(D) = {a, s}γ_{2}(D), proving the second part of Theorem 2.1.

Remarks. 1) The assertion that γ_{3}(D) is a symbol was also proven in [BM] (Th. 2.10) by quite a different method.

2) If the division algebra corresponding to D_{F (\sqrt{a})} is defined over F, then Theorem 2.1 becomes trivial. Indeed, in this case D ≃ (a, s) ⊗_{F} (b, r) ⊗_{F} (c, r) for some b, c, r, s, t ∈ F^{∗}, hence γ_{3}(D) = {a, b, c, s, r, t} and {a}γ_{2}(D) = {a, b, c, r, t}. However, in general an algebra of exponent 2 and index 8 is not a tensor product of three quaternion algebras ([ART], [Ka], [S1]).

3) It is easy to see that modulo Lemma 2.4 Theorem 2.1 is equivalent to the equation {a, c, s}D = γ_{2}(D){c}, which looks similar to the equation in Lemma 2.3. However, we do not know any direct proof of it.

In the following corollary of Theorem 2.1 we give a necessary condition for the existence of subfields of D of certain type (here we consider D as a division algebra).

Corollary 2.7. Let F be a field, char F ≠ 2, \sqrt{−1} ∈ F^{∗}, D ∈_{2} Br F, ind D = 8. Then

1) Suppose that there exist α, β, a ∈ F such that ind D_{F (\sqrt{(α+β\sqrt{a})^{2}−\sqrt{a}})} = 2.

(In other words, D contains the field F (\sqrt{(α + β\sqrt{a})^{2} − \sqrt{a}}). Then γ_{3}(D) = 0.

2) Suppose that L/F is a field extension of degree 8, decomposing into a tower of two cyclic extensions, such that D_{L} = 0. Then γ_{3}(D) = 0.

Proof. 1) Obviously, ind D_{F (\sqrt{a})} = 4 and γ_{2}(D)_{F (\sqrt{(α+β\sqrt{a})^{2}−\sqrt{a}})} = 0. Hence

γ_{2}(D)_{F (\sqrt{a})} = \{(α + β\sqrt{a})^{2} − \sqrt{a}, ω_{1}, ω_{2}, ω_{3}\} for some ω_{1}, ω_{2}, ω_{3} ∈ F (\sqrt{a}). This implies

aγ_{2}(D) = N_{F (\sqrt{a})/F} \{\sqrt{a}, (α + β\sqrt{a})^{2} − \sqrt{a}, ω_{1}, ω_{2}, ω_{3}\} = N_{F (\sqrt{a})/F} (0) = 0,

since \{\sqrt{a}, (α + β\sqrt{a})^{2} − \sqrt{a}\}=0. By Theorem 2.1 we are done.

2) If K = F (\sqrt[4]{a}) and L/K is a quadratic extension, we are done by part 1), setting α = β = 0. If a ∈ F^{∗} and L/F (\sqrt{a}) is a cyclic extension of degree 4, then ind(D)_{F (\sqrt{a})} = 4. Moreover, by [RST] or Theorem 1.1 we have γ_{2}(D)_{F (\sqrt{a})} = γ_{2}(D_{F (\sqrt{a})}) = 0. Hence

\{a\}γ_{2}(D) = N_{F (\sqrt{a})}(\{\sqrt{a}\}γ_{2}(D)_{F (\sqrt{a})}) = N_{F (\sqrt{a})}(0) = 0.

Again the proof is finished by Theorem 2.1. □

Corollary 2.8. Under the previous notation suppose that D is a crossed product for a group G different from (ℤ/2ℤ)^{3}. Then γ_{3}(D) = 0.

Proof. We will give two proofs of this statement. The first one is based on the obvious observation that G contains an element, say σ, of order 4. Let L/F be a subfield of D with the Galois group G and K = G^{<σ>}. Then the extensions K/F and L/K are cyclic, so we are done by Corollary 2.7.

The second proof was proposed by the referee. As was mentioned in the proof of Theorem 2.1, the quadratic form q_{D}(x) = Trd_{D}(x^{2}) is similar to the 6-fold Pfister form, corresponding to γ_{3}(D) under the isomorphism k_{6}(F) ≃ I^{6}(F)/I^{7}(F) ([BM], Th. 2.10). Hence it suffices to prove that q_{D} is isotropic. To do this choose as in the first proof an element σ ∈ G of order 4, and l ∈ L^{∗} such that σ^{2}(l) = −l. There exists x_{σ} ∈ D^{∗} such that x_{σ}l′ = σ(l′)x_{σ} for any l′ ∈ L. It follows that l^{−1}x_{σ}^{2}l = −x_{σ}^{2}, hence

q_{D}(x_{σ}) = Trd(x_{σ}^{2}) = Trd(l^{−1}x_{σ}^{2}l) = Trd(−x_{σ}^{2}) = −q_{D}(x_{σ}),

which implies q_{D}(x_{σ}) = 0. □

Finally we pose a few related problems, which seem to be interesting:

Open questions.

1) Can one extend Theorem 1 to cyclic elements of arbitrary exponent ?

2) Suppose exp α = p, ind α = p^{n}. Is it true that γ_{i}(α) = 0 if i > n ? (Even in the case p = 2 the answer is unknown).

3) Given an odd prime number p, are the conditions ξ_{p} ∈ F and {a, b, c, d} = 0 ∈ K_{4}(F)/p sufficient for the algebra (a, b)_{p} ⊗_{F} (c, d)_{p} to be cyclic ?

4) Suppose that D is a division algebra, whose image in the Brauer group is cyclic. Is D a cyclic algebra ? (In view of Corollary 1.4 the answer is positive for biquaternion algebras if all 2-primary roots of unity are contained in F).

5) Suppose that exp D = 2, ind D = 8, γ_{3}(D) = 0. Is {a}γ_{2}(D) = 0 for some a ∈ F^{∗} such that ind D_{F (\sqrt{a})} = 4 ?

Acknowledgement. I am grateful to Professor A.S. Merkurjev for very useful discussion concerning the questions considered in this paper.

REFERENCES

[ART] Amitsur S.A., Rowen L.H. and Tignol J.-P., Division algebras of degree 4 and 8 with involution, Israel J. Math. 33 (1979), 133-148.

[BM] Baek S., Merkurjev A.S., Invariants of simple algebras, Manuscripta Math. 129 (2009), 409-421.

[EL] Elman R., Lam T.Y., Quadratic forms under algebraic extensions, Math. Ann. 219 (1976), 21-42.

[ELW] Elman R., Lam T.Y., Wadsworth A.R., Amenable fields and Pfister extensions, Queen’s Papers Pure Appl. Math. 46 (1976), 445-492.

[GS] Gille P., Szamuely T., Central simple algebras and Galois cohomology, Cambridge studies in advanced mathematics 101 (2006).

[K] Kahn B., Comparison of some field invariants, J. Algebra 220 (2000), no. 2, 485-492.

[Ka] Karpenko N.A., Codimension 2 cycles on Severi-Brauer varieties, K-Theory 13 (1998), 305-330.

[MS] Merkurjev A.S., Suslin A.A., K-cohomology of Severi-Brauer varieties and the norm residue homomorphism (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011-1046.

[P] Pierce R., Associative algebras, Graduates texts in mathematics, Springer-Verlag New York Inc. 88 (1982).

[RST] Rost M., Serre J.-P., Tignol J.-P., La forme trace d’une algebre simple centrale de degre 4, C.R. Acad. Sci. Paris, Ser. I 342 (2006), no. 2, 83-87.

[S1] Sivatski A.S., On indecomposable algebras of exponent 2, Israel J. Math 164 (2008), 365-379.

[S2] Sivatski A.S., Nonexcellence of certain field extensions, Journal of Mathematical Sciences 145 (2007), no. 1, 4811-4817.

[Vi] Vial C., Operations in Milnor K-theory, J. Pure Appl. Algebra 213 (2009), no. 7, 1325-1345.

[V] Voevodsky V., On motivic cohomology with ℤ/l- coefficients, Ann. of Math. 174 (2) (2011), no. 1, 401-438.

[W] Weibel C., The norm residue isomorphism theorem, Journal of Topology 2 (2009), no. 2, 346-372.

e-mail: alexander.sivatski @gmail.com