TRIPLE LINKAGE

KARIM JOHANNES BECHER

Abstract. For a field the condition is studied that any triple of (bilinear) Pfister forms of a given dimension are linked. This is a strengthening of the condition of linkage investigated by Elman and Lam, which asks the same for pairs of Pfister forms. In characteristic different from two this condition for triples of 2-fold Pfister forms is related to the Hasse number. Classification (MSC 2010): 11E04, 11E81, 19D45 Keywords: field, Milnor K-theory, symbol, linkage, bilinear Pfister form, quadratic form, Hasse number, u-invariant

1. Introduction

Milnor’s seminal article [9] on K-theory of fields had an enormous impact on quadratic form theory. In a series of articles Elman and Lam explored the correspondence between Pfister forms and symbols (canoncial generators) in the K-theory modulo 2 of a field. The notion of linkage for Pfister forms was introduced in [4]. With the definition from [4, Sect. 4] one can consider linkage of a finite number of Pfister forms. However, the study of linkage has mostly been limited to pairs of Pfister forms. Initially, this study was restricted to fields of characteristic different from 2, where quadratic forms are characterised by their associated (symmetric bilinear) polar forms.

When trying to extend notions and statements to cover the case of characteristic 2, one has to choose between quadratic forms or symmetric bilinear forms. In this article we work mainly in the setup of Milnor K-theory over a field of arbitrary characteristic. We study linkage of symbols in the Milnor K-groups modulo 2, or equivalently, of symmetric bilinear Pfister forms. In particular we study the condition that a certain Milnor K-group modulo 2 has triple linkage, i.e. that any three symbols have a common linkage. This condition turns out to have stronger consequences than usual linkage, in particular on the vanishing of higher K-groups. In the last section we focus on fields of characteristic different from 2 and relate the condition of triple linkage to quadratic forms and the Hasse number ũ (the u-invariant if the field is nonreal).

For a recent study of triple linkage of quadratic Pfister forms covering fields of characteristic 2 we refer the reader to [1].

Date: 9 October, 2017.

2. Symbols and Pfister forms

We refer to [3] for standard results used from quadratic form theory. Let E be a field. By a form over E we mean a pair (V, b) of a finite-dimensional E-vector space V and a non-degenerate symmetric bilinear form b on V. We use an equality sign to indicate that two forms are isometric.

Let n always denote a nonnegative integer. Given a_{1}, …, a_{n} ∈ E^{×} we denote the bilinear n-fold Pfister form ⟨1, −a_{1}⟩⊗⋯⊗⟨1, −a_{n}⟩ over E by ⟨⟨a_{1}, …, a_{n}⟩⟩. In the sequel we refer to bilinear Pfister forms simply as Pfister forms. Given a Pfister form π, the orthogonal complement of the subform ⟨1⟩ in π is called the pure part of π.

Theorem 2.1 (Elman-Lam). Let r ∈ ℕ. Let ρ be an anisotropic r-fold Pfister form over E and let ρ′ denote its pure part. Let π be a Pfister form over E such that π ⊗ ρ is anisotropic and let c_{1} ∈ E^{×} be such that −c_{1} is represented by π ⊗ ρ′. Then there exist c_{2}, …, c_{r} ∈ E^{×} such that π ⊗ ρ ≃ π ⊗ ⟨⟨c_{1}, …, c_{r}⟩⟩.

Proof: See [3, Proposition 6.15] or [4, Theorem 2.6]. □

We denote by k_{n}E the nth Milnor K-group of E modulo 2; this is the abelian group generated by symbols {a_{1}, …, a_{n}} with a_{1}, …, a_{n} ∈ E^{×} which are subject to the defining relations that the map (E^{×})^{n} ⟶ k_{n}E given by (a_{1}, …, a_{n}) ↦ {a_{1}, …, a_{n}} is multilinear and further that {a_{1}, …, a_{n}} = 0 whenever a_{i} ∈ E^{×2} for some i ⩽ n or a_{i} + a_{i+1} = 1 for some i < n. The direct sum ⨁_{n∈ℕ} k_{n}E is a graded ring with the multiplication induced by concatenation of symbols.

We recall some results from [3] on the relation of symbols and Pfister forms, which for fields of characteristic different from 2 go back to [4]. We begin with the one-to-one correspondence between symbols and Pfister forms.

Theorem 2.2 (Elman-Lam). For a_{1} …, a_{n}, b_{1}, …, b_{n} ∈ E^{×}, we have

{a_{1}, …, a_{n}} = {b_{1}, …, b_{n}} if and only if ⟨⟨a_{1}, …, a_{n}⟩⟩ = ⟨⟨b_{1}, …, b_{n}⟩⟩ .

Proof: See [3, Theorem 6.20]. □

We denote by ΣE^{2} the subgroup of E^{×} consisting of the non-zero sums of squares in E. Recall that the field E is real if −1 ∉ ΣE^{2}, nonreal otherwise. For m ∈ ℕ we denote by D_{E}(m) the subset of ΣE^{2} consisting of the elements that are sums of m squares in E.

Corollary 2.3. For a ∈ E^{×} and the symbol τ = {−1, …, −1} in k_{n}E the following hold:

(a) a ∈ D_{E}(2^{n}) if and only if τ · {a} = 0 in k_{n+1}E.

(b) a ∈ D_{E}(2^{n}−1) if and only if τ = {a, a_{2}, …, a_{n}} for certain a_{2}, …, a_{n} ∈ E^{×}.

Proof: This follows from Theorem 2.1 and Theorem 2.2. □

Lemma 2.4. Let τ and τ′ be symbols in k_{n}E and let a, a′ ∈ E^{×} be such that τ · {a} = τ′ · {a′}. Then τ · {a} = τ · {c} = τ′ · {c} = τ′ · {a′} for some c ∈ E^{×}.

Proof: This follows from [3, Corollary 6.16 and Theorem 6.20]. □

3. Linkage

Assume in the sequel that n ⩾ 2. Given two symbols σ_{1}, σ_{2} ∈ k_{n}E, the sum σ_{1}+σ_{2} ∈ k_{n}E is equal to a symbol if and only if there exists a symbol σ′ ∈ k_{n−1}E and b_{1}, b_{2} ∈ E^{×} such that σ_{i} = σ′ · {b_{i}} for i = 1, 2 (see [4, Lemma 5.4]); in this case we say that σ_{1} and σ_{2} are linked.

We say that k_{n}E is linked if any two symbols in k_{n}E are linked (which in the terminology of [6] corresponds to saying that I^{n}E is linked.) Obviously, if k_{n}E is linked, then so is k_{m}E for any integer m ⩾ n.

The following statement was obtained in [6, Corollary 2.8 and Corollary 2.9]. For convenience of the reader we include a compact proof, whose first lines follow [5, Sect. 3, Example 3]. The statement should be compared with Theorem 5.1.

Theorem 3.1 (Elman-Lam). Assume that k_{n}E is linked. Then we have that ΣE^{2} = D_{E}(2^{n+1}) and k_{n+2}E = {−1, …, −1} · k_{1}E. In particular, if E is nonreal then k_{n+2}E = 0.

Proof: Consider an arbitrary symbol τ ∈ k_{n−2}E and a_{1}, a_{2}, b_{1}, b_{2} ∈ E^{×}. Since k_{n}E is linked and by Lemma 2.4, there exist c_{1}, c_{2} ∈ E^{×} and a symbol σ ∈ k_{n−1}E such that τ · {a_{i}, b_{i}} = τ · {a_{i}, c_{i}} = σ · {c_{i}} for i = 1, 2. It follows that τ · {a_{1}, c_{1}, c_{2}} = σ · {c_{1}, c_{2}} = τ · {a_{2}, c_{1}, c_{2}}, whereby

τ · {a_{1}, b_{1}, a_{2}, b_{2}} = τ · {a_{1}, c_{1}, a_{2}, c_{2}} = τ · {a_{2}, c_{1}, a_{2}, c_{2}} = {−1, c_{1}} · τ · {a_{2}, b_{2}} .

This argument shows that for any a_{1}, …, a_{n+2} ∈ E^{×} there exists c ∈ E^{×} such that

{a_{1}, …, a_{n+2}} = {a_{2}, …, a_{n+1}, −1, c} .

Applying this rule n + 1 times, we conclude that every symbol in k_{n+2}E is of the form {−1, …, −1, c} with c ∈ E^{×}. Hence k_{n+2}E = {−1, …, −1} · k_{1}E.

Moreover, if a_{1}, …, a_{n+2} ∈ E^{×} are such that a_{2} ∈ D_{E}(2), then {−1, a_{2}} = 0 and we obtain from the above rule that {a_{1}, …, a_{n+2}} = 0. Hence we have {a} · k_{n+1}E = 0 for any a ∈ D_{E}(2).

Consider an element c ∈ D_{E}(2^{n+1} + 1). We write c = a + b with a ∈ D_{E}(2) and b ∈ D_{E}(2^{n+1} − 1). In k_{n+1}E we obtain that {−1, …, −1} = {−b} · τ for a symbol τ in k_{n}E, by Corollary 2.3. Since c − b = a we have {−b, c} = {a, bc}. As {a} · k_{n+1}E = 0 we obtain in k_{n+2}E that

{−1, …, −1, c} = {−b, c} · τ = {a, bc} · τ = 0 ,

which shows that c ∈ D_{E}(2^{n+1}). This argument shows that ΣE^{2} = D_{E}(2^{n+1}).

Assume finally that E is nonreal. If −1 ∈ E^{×2} then {−1} = {1} = 0 in k_{1}E. If char(E) ≠ 2 then E^{×} = ΣE^{2} = D_{E}(2^{n+1}). Hence in any case we obtain that k_{n+2}E = {−1, …, −1} · k_{1}E = 0. □

If E is nonreal and k_{n}E is linked where n ⩾ 2, then k_{n+2}E vanishes by Theorem 3.1, but we may have that k_{n+1}E ≠ 0, as the following well-known example shows.

Example 3.2. For E = ℂ((t_{1})) … ((t_{n+1})), k_{n}E is linked and k_{n+1}E ≃ ℤ/2ℤ.

4. The linkage pairing

We are going to investigate an operation on linked symbols. Let n ⩾ 2. To any pair of linked symbols in k_{n}E we associate a symbol in k_{n+1}E.

Proposition 4.1. Let σ_{1}, σ_{2} ∈ k_{n}E be two linked symbols. There is a unique symbol ρ ∈ k_{n+1}E such that, for any symbol τ ∈ k_{n−1}E and any a_{1}, a_{2} ∈ E^{×} with σ_{i} = τ · {a_{i}} for i = 1, 2, we have ρ = τ · {a_{1}, a_{2}}.

Proof: By the hypothesis there exist a symbol τ ∈ k_{n−1}E and two elements a_{1}, a_{2} ∈ E^{×} with σ_{i} = τ · {a_{i}} for i = 1, 2. Suppose we have another symbol τ′ ∈ k_{n−1}E and a′_{1}, a′_{2} ∈ E^{×} with σ_{i} = τ′ · {a′_{i}} for i = 1, 2. By Lemma 2.4 there exist c_{1}, c_{2} ∈ E^{×} such that τ · {c_{i}} = σ_{i} = τ′ · {c_{i}} holds for i = 1, 2. We obtain that τ · {a_{1}, a_{2}} = τ · {c_{1}, c_{2}} = τ′ · {c_{1}, c_{2}} = τ′ · {a′_{1}, a′_{2}}. □

Corollary 4.2. Let a, b, c ∈ E^{×} and let τ be a symbol in k_{n−1}E such that τ · {−1, a} = 0. Assume that there exist a symbol ρ in k_{n}E and x, y, z ∈ E^{×} such that τ · {a, b} = ρ · {x}, τ · {a, c} = ρ · {y} and τ · {b, c} = ρ · {z}. Then τ · {a, b, c} = 0.

Proof: By Proposition 4.1 we obtain that ρ · {x, z} = τ · {a, b, c} = ρ · {y, z}, whereby ρ · {xy, z} = 0. We have τ · {b, −bc} = τ · {b, c} = ρ · {z}. Since τ · {−1, a} = 0 we further have τ · {a, −bc} = τ · {a, bc} = ρ · {xy}. We conclude with Proposition 4.1 that τ · {a, b, c} = τ · {a, b, −bc} = ρ · {xy, z} = 0. □

Corollary 4.3. Assume that k_{n}E is linked. We obtain a surjective pairing

⟨·, ·⟩ : k_{n}E × k_{n}E ⟶ k_{n+1}E

by letting ⟨τ · {a_{1}}, τ · {a_{2}}⟩ = τ · {a_{1}, a_{2}} for any symbol τ ∈ k_{n−1}E and any a_{1}, a_{2} ∈ E^{×}.

Proof: Let σ_{1}, σ_{2} ∈ k_{n}E be given. As k_{n}E is linked, there exist a_{1}, a_{2} ∈ E^{×} and a symbol τ ∈ k_{n−1}E such that σ_{i} = τ · {a_{i}} for i = 1, 2. By Proposition 4.1 the symbol ρ = τ · {a_{1}, a_{2}} ∈ k_{n+1}E only depends on σ_{1} and σ_{2} but not on the choice of τ and a_{1}, a_{2} ∈ E^{×}. Hence, the pairing is well-defined. As k_{n}E is linked, so is k_{n+1}E, and it follows that the pairing is surjective. □

If k_{n}E is linked then we call the pairing in Corollary 4.3 the linkage pairing on k_{n}E.

Theorem 4.4. Assume that k_{n}E is linked. Then the following are equivalent: (i) The linkage pairing on k_{n}E is bilinear. (ii) ΣE^{2} = D_{E}(2^{n}) and k_{n+1}E = {−1, …, −1} · k_{1}E. (iii) Either k_{n+1}E = 0, or E is real and the rule c ↦ {−1, …, −1, c} deter-

mines an isomorphism E^{×}/ ΣE^{2} ⟶ k_{n+1}E.

Proof: As a consequence of the definition of the linkage pairing

⟨·, ·⟩ : k_{n}E × k_{n}E ⟶ k_{n+1}E

we have for any ρ, ρ′ ∈ k_{n}E that ⟨ρ, ρ′⟩ = ρ · {d} for some d ∈ E^{×}.

(i ⇒ ii) Consider an arbitrary symbol τ ∈ k_{n−2}E and a, b, c ∈ E^{×}. Set σ_{1} = {a, b}, σ_{2} = {a, c}, σ_{3} = {b, c} and σ_{4} = {−ab, c}. We obtain that ⟨τ · σ_{i}, τ · σ_{1}⟩ = τ · {a, b, c} for i = 2, 3, 4. Assuming that the pairing is bilinear, we get that

⟨τ · (σ_{2} + σ_{3} + σ_{4}), τ · σ_{1}⟩ = τ · {a, b, c} .

Since σ_{2} + σ_{3} + σ_{4} = {−1, c}, we conclude that

τ · {a, b, c} = ⟨τ · {−1, c}, τ · {a, b}⟩ = τ · {−1, c, d}

for some d ∈ E^{×}. This argument shows that, for any c_{1}, …, c_{n}, c_{n+1} ∈ E^{×}, there exists d ∈ E^{×} such that {c_{1}, …, c_{n}, c_{n+1}} = {−1, c_{1}, …, c_{n−1}, d} in k_{n+1}E. Using this rule n times we obtain for any c_{1}, …, c_{n}, c_{n+1} ∈ E^{×} that

{c_{1}, …, c_{n+1}} = {−1, …, −1, c_{1}, d} = {−1, …, −1, d′}

for some d, d′ ∈ E^{×}. This shows that k_{n+1}E = {−1, …, −1} · k_{1}E and that {a} · k_{n}E = 0 for any a ∈ D_{E}(2^{n−1}).

Consider now an element c ∈ D_{E}(2^{n} + 1). We choose a, b ∈ D_{E}(2^{n−1}) such that c − a − b is a square in E. Then we have {−a, −b, c} = 0 in k_{3}E, whereby {a, −b, c} = {−1, −b, c}. For the symbol ϵ = {−1, …, −1} in k_{n−2}E we obtain that ϵ · {−1, b} = 0 and conclude that

ϵ · {−1, −1, c} = ϵ · {−1, −b, c} = ϵ · {a, −b, c} = 0 ,

for a ∈ D_{E}(2^{n−1}). Hence {−1, …, −1, c} = 0 in k_{n+1}E, whereby c ∈ D_{E}(2^{n}). This shows that ΣE^{2} = D_{E}(2^{n}).

(ii ⇒ iii) This implication is obvious.

(iii ⇒ i) Let ε = {−1, …, −1} in k_{n−1}E. For any symbol τ ∈ k_{n−1}E and any a, b ∈ E^{×} we have τ · {a} · τ · {b} = ε · τ · {a, b} = ε · ⟨τ · {a}, τ · {b}⟩. Since k_{n}E is linked, this means that

ρ · ρ′ = ε · ⟨ρ, ρ′⟩ in k_{2n}E for any ρ, ρ′ ∈ k_{n}E .

Hence, the pairing k_{n}E × k_{n}E ⟶ k_{2n}E, (ρ, ρ′) ↦ ε · ⟨ρ, ρ′⟩ is bilinear. On the other hand, (iii) implies that k_{n+1}E ⟶ k_{2n}E, ξ ↦ ε · ξ is an isomorphism. Therefore the pairing ⟨·, ·⟩ is bilinear. □

5. Triple linkage

Let n ⩾ 2. We say that k_{n}E has triple linkage if for any three symbols σ_{1}, σ_{2}, σ_{3} ∈ k_{n}E there exist a symbol τ ∈ k_{n−1}E and a_{1}, a_{2}, a_{3} ∈ E^{×} such that σ_{i} = τ · {a_{i}} for i = 1, 2, 3. Note that this implies that k_{n}E is linked.

Theorem 5.1. Assume that k_{n}E has triple linkage. Then ΣE^{2} = D_{E}(2^{n}) and k_{n+1}E = {−1, …, −1} · k_{1}E. In particular, if E is nonreal then k_{n+1}E = 0.

Proof: Let ⟨·, ·⟩ : k_{n}E × k_{n}E ⟶ k_{n+1}E be the linkage pairing. Consider three symbols σ_{1}, σ_{2}, σ_{3} ∈ k_{n}E. By the hypothesis there exist a symbol τ ∈ k_{n−1}E and a_{1}, a_{2}, a_{3} ∈ E^{×} such that σ_{i} = τ ·{a_{i}} for i = 1, 2, 3. As σ_{1} +σ_{2} = τ ·{a_{1}a_{2}} we obtain that

⟨σ_{1} + σ_{2}, σ_{3}⟩ = τ · {a_{1}a_{2}, a_{3}} = τ · {a_{1}, a_{3}} + τ · {a_{2}, a_{3}} = ⟨σ_{1}, σ_{3}⟩ + ⟨σ_{2}, σ_{3}⟩ .

Hence the linkage pairing is bilinear and Theorem 4.4 yields the statement. □

Question 5.2. If k_{n}E has triple linkage, does then any finite number of symbols in k_{n}E have a common linkage (by a symbol in k_{n−1}E)?

Triple linkage holds for k_{n}E if E is a C_{n}-field, in the terms of Tsen-Lang Theory (see [11, Chap. 5]). This is a direct consequence of the next statement. For n = 1 and |S| = 3 the statement corresponds to [13, Proposition 9].

Proposition 5.3. Assume that there exists a finite system S of nonzero symbols in k_{n}E that do not have a common linkage. Then there exists an anisotropic system of |S| − 1 quadratic forms in |S| · 3 · 2^{n−2} variables over E.

Proof: Let m ∈ ℕ be as large as possible such that there exist a_{1}, …, a_{m} ∈ E^{×} for which the symbol {a_{1}, …, a_{m}} factors every element of S. By the hypothesis m ⩽ n − 2. We set π = ⟨⟨a_{1}, …, a_{m}⟩⟩. Using Theorem 2.1 and the one-to-one correspondence between Pfister forms and symbols, we choose for σ ∈ S an (n − m)-fold Pfister form ρ_{σ} over E such that σ corresponds to the n-fold Pfister form π ⊗ ρ_{σ} over E and we denote by ρ′_{σ} the pure part of ρ_{σ}. Note that dim(π⊗ρ′_{σ}) = 2^{n}−2^{m} for any σ ∈ S. By Theorem 2.1 and by the maximality of m, there exists no element c ∈ E^{×} such that −c is represented by all the forms π ⊗ ρ′_{σ} with σ ∈ S. We fix σ_{0} ∈ S and set S′ = S ∖ {σ_{0}}. Considering each of the forms π ⊗ ρ′_{σ} for σ ∈ S with its own variables, we obtain an anisotropic system of quadratic forms (π ⊗ ρ′_{σ_{0}} − π ⊗ ρ′_{σ})_{σ∈S′} in |S| · (2^{n} − 2^{m}) variables over E. If m < n − 2 we may substitute zero for some of these variables. So in any case we obtain an anisotropic system of |S| − 1 quadratic forms over E in exactly |S| · 3 · 2^{n−2} variables. □

Let ũ(E) denote the Hasse number of E, which is defined as the supremum in ℕ ∪ {∞} on the dimension of anisotropic totally indefinite quadratic forms over E (cf. [11, Chap. 8, Sect. 3]). The study of this invariant was initiated in [7], the notation was introduced in [2]. The definition of the Hasse number captures one of several possibilities to study bounds on the dimension of anisotropic quadratic forms in a meaningful way without restriction to nonreal fields. The results below have their main interest in the case where E is nonreal, and in this case ũ(E) is the usual u-invariant (cf. [11, Chap. 8] and [3, Chap. VI]).

Corollary 5.4. If ũ(E(t)) ⩽ 2^{n+1} then k_{n}E has triple linkage.

Proof: Suppose first that E is real and ũ(E(t)) < ∞. It follows by [7, Theorem I] that E is hereditarily euclidean. Hence k_{n}E ≃ ℤ/2ℤ, whereby k_{n}E has triple linkage.

Assume now that E is nonreal and ũ(E(t)) ⩽ 2^{n+1} < 9 · 2^{n−2}. By the Amer-Brumer Theorem [11, Chap. 9, Prop. 1.10], it follows that every pair of quadratic forms in 9 · 2^{n−2} variables over E is isotropic. Hence k_{n}E has triple linkage, by Proposition 5.3. □

The next example shows that the converse to the statement in Corollary 5.4 does not hold.

Example 5.5. Let E_{0} be a quadratically closed field of characteristic not 2 having a finite field extension of even degree. (One can for example take E_{0} as the quadratic closure of ℚ: any polynomial over ℚ having as Galois group a dihedral group of order 2m for an odd positive integer m will have as splitting field over E_{0} an extension of order 2m. See also the discussion of finite extensions of quadratically closed fields [8, Chapter VII, §7].) It follows from this choice of E_{0} that, ũ(E_{0}) = 1 while ũ(F_{0}) ⩾ 2 for some finite separable extension F_{0}/E_{0}. Thus k_{1}F_{0} ≠ 0. Consider the fields of iterated power series in n variables

E = E_{0}((u_{1})) … ((u_{n})) and F = F_{0}((u_{1})) ⋯ ((u_{n})) .

We obtain that ũ(E) = 2^{n} and k_{n+1}F ≠ 0. Since F/E is a finite separable extension, F is the residue field of a discrete valuation on E(t). We conclude that k_{n+2}E(t) ≠ 0. After translation to Pfister forms via [3, Theorem 6.20], the Arason-Pfister Hauptsatz [3, Theorem 6.18] yields that ũ(E(t)) ⩾ 2^{n+2}.

Note that {u_{1}, …, u_{n}} is the only nonzero symbol in k_{n}E. In particular k_{n}E has triple linkage. Finally, k_{n}F does not have triple linkage, for k_{n+1}F ≠ 0.

The following reformulates and enhances [7, Theorem G]. The notation I^{n}E refers to the nth power of the fundamental ideal IE in the Witt ring (of symmetric bilinear forms) of E; recall that I^{n}E is additively generated by the classes of the n-fold Pfister forms over E.

Theorem 5.6 (Elman-Lam-Prestel). Assume that char(E) ≠ 2. The following are equivalent:

(i) ũ(E) ⩽ 4. (ii) E is linked and I^{3}E is torsion-free. (iii) E is linked and the linkage pairing k_{2}E × k_{2}E ⟶ k_{3}E is bilinear. (iv) ΣE^{2} = D_{E}(4) and k_{3}E = {−1, −1} · k_{1}E.

(v) k_{2}E is linked and either k_{3}E = 0, or E is real and c ↦ {−1, −1, c} defines an isomorphism E^{×}/ ΣE^{2} ⟶ k_{3}E.

Proof: Conditions (iii) − (v) are equivalent by Theorem 4.4. The equivalence of (i) and (ii) is shown in [7, Theorem G] and in [2, Theorem 4.7]. By [6, Corollary 2.9] (ii) implies that I^{3}E = 4 · IE. Therefore the equivalence of (ii) and (iv) follows using Theorem 2.2. □

Corollary 5.7. Assume that char(E) ≠ 2 and that k_{2}E has triple linkage. Then ũ(E) ⩽ 4.

Proof: This follows from Theorem 5.1 together with Theorem 5.6. □

In the case where E is nonreal, one can show the converse of Corollary 5.7 by using the following statement, which is a direct consequence of a deep result of Peyre [10, Proposition 6.1] combined with an observation by Sivatski [13, Corollary 11].

Proposition 5.8 (Peyre-Sivatski). Assume that char(E) ≠ 2. Let H be a subgroup of k_{2}E with |H| ⩽ 8. Assume that every element of H is a symbol and that I^{3}E = 0. Then there exists a ∈ E^{×} such that for every σ ∈ H one has σ = {a, b_{σ}} for some b_{σ} ∈ E^{×}.

Proof: By the hypothesis every σ ∈ H corresponds to an E-quaternion algebra. Let σ_{1}, σ_{2}, σ_{3} ∈ k_{2}E be three symbols that generate H and let Q_{1}, Q_{2}, Q_{3} denote the corresponding quaternion algebras. Since I^{3}E = 0 we have that H^{3}(E, ℤ/ℚ(2)) = 0. It follows by [10, Proposition 6.1] that there exists a field extension F/E with [F : E] = 2m for an odd integer m and such that (Q_{i})_{F} is split for i = 1, 2, 3. Then, by [13, Corollary 11], Q_{1}, Q_{2}, Q_{3} have a common slot a ∈ E^{×}. It follows that any σ ∈ H is of the form σ = {a, b_{σ}} with b_{σ} ∈ E^{×}. □

In [13] Sivatski seems to be unaware that he provides a proof of Proposition 5.8. In [13, Corollary 12] he comes to a closely related conclusion, but at the end of his article he asks whether fields of cohomological 2-dimension 2 satisfy the conclusion stated in Proposition 5.8. This was pointed out to the author of the present article by Adam Chapman and David Leep.

Corollary 5.9. Assume that E is nonreal with char(E) ≠ 2. Then ũ(E) ⩽ 4 if and only if k_{2}E has triple linkage.

Proof: One implication is Corollary 5.7, the converse follows from Proposition 5.8. □

The hypothesis in Proposition 5.8 on E can be weakened. Instead of assuming I^{3}E = 0, which requires E to be nonreal, it is sufficient to assume that I^{3}E is torsion-free and that E has the so-called ED-property about field orderings introduced in [12]. This can be proven by using algebras with involutions and skew-hermitian forms over quaternion algebras. In this way Proposition 5.8 is recovered with a different proof, which in particular is independent of the algebraic geometry behind Peyre’s result [10, Proposition 6.1].

Since ũ(E) < ∞ implies that E satisfies the ED-property, the generalisation carries over to Corollary 5.9 and makes the condition that E is nonreal superfluous: If char(E) ≠ 2, then ũ(E) ⩽ 4 if and only if k_{2}E has triple linkage.

The author is planning to give details on this argument in a future article.

Acknowledgments. This work was supported by the FWO Odysseus Programme (project Explicit Methods in Quadratic Form Theory), funded by the Fonds Wetenschappelijk Onderzoek – Vlaanderen. I further wish express my gratitude to Andrew Dolphin, Parul Gupta, David Leep and Anne Quéguiner-Mathieu for inspiring discussions around this topic. I thank Adam Chapman and David Leep in particular for pointing out Proposition 5.8. Finally, I am grateful to the referee for helping to improve the readability of the article.

References

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[9] J.W. Milnor. Algebraic K-theory and quadratic forms. Inventiones Math. 9 (1970): 318– 344.

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[11] A. Pfister. Quadratic Forms with Applications to Algebraic Geometry and Topology. LMS Lecture Notes series, 217. Cambridge University Press. Cambridge, 1995.

[12] A. Prestel, R. Ware. Almost isotropic forms. J. Lond. Math. Soc. Ser. II 19 (1979): 241–244.

[13] A.S. Sivatski. Linked triples of quaternion algebras. Pacific J. Math. 268 (2014): 465–476.

Departement Wiskunde–Informatica, Universiteit Antwerpen, Belgium

E-mail address: KarimJohannes.Becher@uantwerpen.ie