BIQUATERNION DIVISION ALGEBRAS OVER RATIONAL FUNCTION FIELDS

KARIM JOHANNES BECHER

Abstract. Let E be a field which is the center of a quaternion division algebra and which is not real euclidean. Then there exists a biquaternion division algebra over the rational function field E(t) which does not contain any quaternion algebra defined over E. The proof is based on the study of Bezoutian forms developed in [1]. Keywords: Milnor K-theory, quadratic form, valuation, ramification, Bezoutian form Classification (MSC 2010): 12E30, 12G05, 12Y05, 19D45

1. Introduction

Let E be a field of characteristic different from 2. Let E(t) denote the rational function field over E, where t is an indeterminate. It is easy to show that there exists a quaternion division algebra over E(t) if and only if E has some field extension of even degree. The purpose of this article is to provide sufficient conditions to have that there exists a biquaternion division algebra over E(t).

In [6] and [4, Sections 3 and 4] such examples were given in the case where E is a local number field. In this article a different approach will be presented that uses the study of ramification sequences via associated Bezoutian forms developped in [1].

For a, b ∈ E^{×} consider the biquaternion algebra

B = (t^{2} + (a + 1)t + a, a) ⊗_{E(t)} (t^{2} + at + a, ab)

over E(t). We will obtain by Theorem 4.2 that, for having that B is a division algebra, it suffices that the E-quaternion algebra (a, b) is non-split and that ab, (a − 4)b ∉ E^{×2}. More precisely, under the same conditions the ramification of B (with respect to the valuations on E(t) which are trivial on E) differs from the ramification of any quaternion algebra over E(t), which means that B has Faddeev index 4, in the terminology of [4].

This will yield the following sufficient condition for the existence of biquaternion division algebras over E(t).

Date: October 10, 2017.

Theorem. Assume that E is not real euclidean and that there exists a non-split quaternion algebra over E. Then there exists a biquaternion division algebra over E(t) which does not contain any quaternion algebra defined over E.

Recall that E is real euclidean if the set of squares in E is an ordering of E. It is clear that this case has to be excluded in the Theorem. In fact, if E is real euclidean, then every E(t)-quaternion algebra is of the form (−1, f) for some f ∈ E[t] and it follows that every E(t)-biquaternion algebra has zero divisors.

As a consequence of the Theorem one may therefore notice that whenever there exist biquaternion division algebras over E(t) then there exist also biquaternion algebras over E(t) of Faddeev index 4.

Note finally that the converse of the Theorem does not hold. It was shown in [2] that one can construct a field E of cohomological dimension 1 – hence in particular such that every E-quaternion algebra is split – and such that there exist biquaternion division algebras over E(t).

2. Preliminaries

For an E-algebra A and a field extension F/E, we denote by A_{F} the F -algebra A ⊗_{E} F. Recall that an E-algebra A is central simple if and only if A_{F} ≃ 𝕄_{n}(F) for some field extension F/E and a positive integer n; we say that A is split if one can take F = E, that is if A ≃ 𝕄_{n}(E). Note that any central simple algebra is finite-dimensional and in particular it either has zero divisors or it is a division algebra.

An E-quaternion algebra is a 4-dimensional central simple E-algebra. For a, b ∈ E^{×} an E-quaternion algebra denoted (a, b)_{E} or just (a, b) is obtained by endowing the vector space

E ⊕ Ei ⊕ Ej ⊕ Ek

with the multiplication given by the rules i^{2} = a, j^{2} = b and ij = k = −ji. Any E-quaternion algebra is isomorphic to (a, b)_{E} for certain a, b ∈ E^{×}. A quaternion algebra is either split or it is a division algebra.

An E-biquaternion algebra is an E-algebra which is isomorphic to Q ⊗_{E} Q′ for two E-quaternion algebras Q and Q′. In particular, biquaternion algebras are central simple. Given an E-biquaternion algebra B and an E-quaternion subalgebra Q of B, we can decompose B ≃ Q ⊗_{E} Q′ with the E-quaternion algebra Q′ given as the centralizer of Q in B.

For our analysis of quaternion and biquaternion algebras over E and E(t), we will work in the second Milnor K-groups modulo 2 of the fields.

For n ∈ ℕ 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. Note that k_{0}E ≃ ℤ/2ℤ and k_{1}E ≃ E^{×}/E^{×2}. Here we only consider k_{n}E (and k_{n}E(t)) for n = 1, 2. The group k_{2}E is in tight relation to the Brauer group.

We denote by Br(E) the Brauer group of E and by Br_{2}(E) its 2-torsion part. Recall that there is a unique homomorphism

k_{2}E → Br_{2}(E)

that sends any symbol {a, b} with a, b ∈ E^{×} to the Brauer equivalence class of the E-quaternion algebra (a, b)_{E}. Merkurjev’s Theorem asserts that this is in fact an isomorphism. We only need special instances of this statement. For a, b ∈ E^{×} we have {a, b} = 0 in k_{2}E if and only if the E-quaternion algebra (a, b)_{E} is split. Furthermore, for a, b, c, d ∈ E^{×} the E-biquaternion algebra (a, b)⊗_{E}(c, d) has zero divisors if and only if {a, b} + {c, d} = {e, f} for certain e, f ∈ E^{×}. These two facts can be proven by elementary means, without using Merkurjev’s Theorem.

To the study of k_{2}E(t) one uses Milnor’s Exact Sequence (2.1). To explain it we first need to define the tame symbol map ∂_{v} with respect to a ℤ-valuation v.

Let F be a field. By a ℤ-valuation on F we mean a valuation with value group ℤ. Given a ℤ-valuation v on F we denote by O_{v} its valuation ring and by κ_{v} its residue field. For a ∈ O_{v} let \overline{a} denote the natural image of a in κ_{v}. By [5, (2.1)], for a ℤ-valuation v on F, there is a unique homomorphism ∂_{v} : k_{2}F → k_{1}κ_{v} such that

∂_{v}(\{f, g\}) = v(f) · \{\overline{g}\} in k_{1}κ_{v}

for f ∈ F^{×} and g ∈ O_{v}^{×}. For f, g ∈ E^{×} we obtain that f^{−v(g)}g^{v(f)} ∈ O_{v}^{×} and

∂_{v}(\{f, g\}) = \{(−1)^{v(f)v(g)}\overline{f^{−v(g)}g^{v(f)}}\} in k_{1}κ_{v} .

We turn to the case where F = E(t). Let P denote the set of monic irreducible polynomials in E[t]. Any p ∈ P determines a ℤ-valuation v_{p} on E(t) which is trivial on E and with v_{p}(p) = 1. There is further a unique ℤ-valuation v_{∞} on E(t) such that v_{∞}(X) = −1. We set P′ = P ∪ {∞}. For p ∈ P′ we write ∂_{p} for ∂_{v_{p}} and we denote by E_{p} the residue field of v_{p}. Note that E_{p} is naturally isomorphic to E[t]/(p) for any p ∈ P and that E_{∞} is naturally isomorphic to E. We call

∂ = ⨁_{p∈P′} ∂_{p} : k_{2}E(t) → ⨁_{p∈P′} k_{1}E_{p}

the ramification map. For p ∈ P′, the norm map of the finite extension E_{p}/E yields a group homomorphism k_{1}E_{p} → k_{1}E. Summation over these maps for all p ∈ P′ yields a homomorphism

N : ⨁_{p∈P′} k_{1}E_{p} → k_{1}E .

Let ℜ_{2}(E) denote the kernel of N. By [3, (7.2.4) and (7.2.5)] we obtain an exact sequence

0 → k_{2}E → k_{2}E(t) →^{∂} ⨁_{p∈P′} k_{1}E_{p} →^{N} k_{1}E → 0 . (2.1)

In particular, ℜ_{2}(E) is equal to the image of ∂. The elements of ℜ_{2}(E) are called ramification sequences.

For a finite set S ⊆ P′ we call ∑_{p∈S}[E_{p} : E] the degree of S and denote it by deg(S). For ρ = (ρ_{p})_{p∈P′} ∈ ⨁_{p∈P′} k_{1}E_{p} we set Supp(ρ) = {p ∈ P′ | ρ_{p} ≠ 0} and abbreviate deg(ρ) = deg(Supp(ρ)), and we call this the support and the degree of ρ. We say that ρ ∈ ℜ_{2}(E) is represented by ξ ∈ k_{2}E(t) if ∂(ξ) = ρ.

3. Bezoutians

We use standard terminology from quadratic form theory. For n ∈ ℕ and a_{1}, …, a_{n} ∈ E^{×} we denote the n-fold Pfister form ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩ by ⟨⟨a_{1}, …, a_{n}⟩⟩. The Witt ring of E is denoted by WE. For a nondegenerate quadratic form φ over E we denote by [φ] its class in WE and we set c·[φ] = [cφ] for c ∈ E^{×}. For c ∈ E^{×} we abbreviate [c] = [⟨c⟩]. For α, α′ ∈ WE we write α ∼ α′ to indicate that α′ = cα for some c ∈ E^{×}.

Consider a square-free polynomial g ∈ E[t] and a polynomial f ∈ E[t] coprime to g. Let θ denote the class of t in E_{g} = E[t]/(g), n = deg(g), and let s_{g} : E_{g} → E be the E-linear form with s_{g}(θ^{i}) = 0 for i = 0, …, n − 2 and s_{g}(θ^{n−1}) = 1. By [1, Proposition 3.1]

q : E_{g} → E, x ↦ s_{g}(f(θ)x^{2})

is a nondegenerate quadratic form over E, called the Bezoutian of f modulo g. We denote the class in WE given by the Bezoutian of f modulo g by

𝔅 (\frac{f}{g} ) .

Bezoutians satisfy some computation rules, which are useful.

3.1. Proposition. For f, g_{1}, g_{2} ∈ E[t] pairwise coprime and with g_{1} and g_{2} monic and square-free, we have

𝔅 (\frac{f}{g_{1}g_{2}} ) = 𝔅 (\frac{fg_{2}}{g_{1}} ) + 𝔅 (\frac{fg_{1}}{g_{2}} ) .

Proof. See [1, Proposition 3.5]. □

3.2. Theorem. Let f, g ∈ E[t] be monic, square-free and coprime. Then

𝔅 (\frac{f}{g} ) + 𝔅 (\frac{g}{f} ) = \{ \begin{matrix}0 & if deg(f) ≡ deg(g) mod 2, \\ [1] & if deg(f) ≢ deg(g) mod 2. \end{matrix}

Proof. See [1, Theorem 3.8]. □

These two rules will be used without explicit mention in the sequel.

3.3. Lemma. Let a_{1}, a_{2} and g_{1}, g_{2} ∈ E[t] monic of even degree and relatively coprime and such that g_{1}t is a square modulo g_{2}. Let f ∈ E[t] be such that a_{i}f is a square modulo g_{i} for i = 1, 2. Then

𝔅 (\frac{f}{g_{1}g_{2}} ) ∼ [⟨⟨a_{1}a_{2}, g_{2}(0)⟩⟩] .

Proof. Let b = g_{2}(0). As g_{1}t is a square modulo g_{2}, we obtain that

𝔅 (\frac{g_{1}}{g_{2}} ) = 𝔅 (\frac{t}{g_{2}} ) = [1] − 𝔅 (\frac{g_{2}}{t} ) = [1] − [b] .

It follows that

𝔅 (\frac{g_{2}}{g_{1}} ) = −𝔅 (\frac{g_{1}}{g_{2}} ) = [b] − [1] .

We obtain that

𝔅 (\frac{f}{g_{1}g_{2}} ) = a_{1}𝔅 (\frac{g_{2}}{g_{1}} ) + a_{2}𝔅 (\frac{g_{1}}{g_{2}} ) = [a_{2}⟨⟨a_{1}a_{2}, b⟩⟩] .

□

4. Ramification sequences not representable by a symbol

In [1, Section 5] Bezoutians are related to ramification sequences and it is shown in [1, Theorem 5.12] that a non-trivial Bezoutian can present an obstruction for the representability of a ramification sequence by a single symbol. Here this will be used to obtain ramification sequences of degree 4 that do not correspond to a symbol.

Given g ∈ E[t] monic and square-free and f ∈ E[t] coprime to g, we denote by

ℜ (\frac{f}{g} )

the element ρ ∈ ℜ_{2}(E) such that ρ_{p} = \{\overline{f}\} for all p ∈ P dividing g and ρ_{p} = 0 for all other p ∈ P; note that ρ_{∞} is given by the condition that ρ ∈ ℜ_{2}(E) = ker(N).

4.1. Proposition. Let g_{1}, g_{2} ∈ E[t] monic of even degree, coprime and such that g_{1}t is a square modulo g_{2}. Let a_{1}, a_{2} ∈ E^{×} be such that the quadratic form ⟨1, −a_{1}a_{2}⟩ over E(t) does not represent g_{2}(0) and for i = 1, 2 one has a_{i} ∉ E_{p}^{×2} for any irreducible factor p of g_{i}. Then

∂ ({g_{1}, a_{1}} + {g_{2}, a_{2}}) ≠ ∂ (σ)

for any symbol σ in k_{2}E(t).

Proof. We set ρ = ∂ ({g_{1}, a_{1}} + {g_{2}, a_{2}}) in ℜ_{2}(E). We have ρ_{∞} = 0 and Supp(ρ) = {p ∈ P | p divides g_{1}g_{2}}. In particular deg(ρ) = deg(g_{1}) + deg(g_{2}), which is even. Suppose there exists a symbol σ in k_{2}E(t) with ∂(σ) = ρ. It follows by [1, Proposition 4.1] that there exist f, g, h ∈ E[t] square-free and pairwise coprime and with g = g_{1}g_{2} such that σ = \{f, gh\}, and ∂(σ) = ℜ (\frac{f}{g} ). By [1, Lemma 4.2] we obtain that

𝔅 (\frac{f}{g} ) = 0 .

Since

ℜ (\frac{f}{g_{1}} ) + ℜ (\frac{f}{g_{2}} ) = ℜ (\frac{f}{g} ) = ∂(σ) = ρ = ℜ (\frac{a_{1}}{g_{1}} ) + ℜ (\frac{a_{2}}{g_{2}} ) ,

we have that a_{i}f is a square modulo g_{i} for i = 1, 2. thus by Lemma 3.3 we get that

[⟨⟨a_{1}a_{2}, g_{2}(0)⟩⟩] ∼ 𝔅 (\frac{f}{g} ) = 0 .

Thus ⟨⟨a_{1}a_{2}, g_{2}(0)⟩⟩ is hyperbolic. Hence g_{2}(0) is represented over E by the quadratic form ⟨1, −a_{1}a_{2}⟩, which contradicts the hypothesis. □

We are ready to prove the statements in terms of symbols which were claimed in the introduction in terms of quaternion algebras. For the translation we rely only the fact that the ramification map ∂ : k_{2}E(t) → ⨁_{p∈P′} k_{1}E_{p} factors over the natural homomorphism k_{2}E(t) → Br_{2}(E(t)).

4.2. Theorem. Let a, b ∈ E^{×} be such that a ∉ E^{×2} and b ∉ aE^{×2} ∪ (a − 4)E^{×2}. The following are equivalent:

(i) {a, b} = 0 in k_{2}E. (ii) There exists a symbol σ ∈ k_{2}E(t) with

∂ ({t^{2} + (a + 1)t + a, a} + {t^{2} + at + a, ab}) = ∂(σ) .

Proof. Set g_{1} = t^{2}+(a+1)t+a, g_{2} = t^{2}+at+a and ρ = ∂ ({g_{1}, a} + {g_{2}, ab}). The polynomials g_{1} and g_{2} are coprime, and we have g_{2}(0) = a and g_{1}t ≡ t^{2} mod g_{2}. The discriminant of g_{2} is a^{2} − 4a. Hence, the conditions on a and b imply that g_{2} is separable and ab is a non-square modulo any irreducible factor of g_{2}. In particular, we have that Supp(ρ) = {p ∈ P | p divides g_{1}g_{2}} and deg(ρ) = 4.

Moreover, as a ∉ E^{×2}, we have that a is a non-square modulo the irreducible factors of g_{1} = (t + 1)(t + a).

If {a, b} ≠ 0, then ⟨1, −b⟩ does not represent a = g_{2}(0), and we conclude by Proposition 4.1 that ρ ≠ ∂(σ) for any symbol σ in k_{2}E(t). Assume conversely that {a, b} = 0. Then ⟨⟨a, b⟩⟩ is hyperbolic. Hence choosing f ∈ E[t] such that f ≡ a mod g_{1} and f ≡ ab mod g_{2}, we obtain by Lemma 3.3 that 𝔅 (\frac{f}{g_{1}g_{2}} ) = 0,

and as ρ = ℜ (\frac{f}{g_{1}g_{2}} ) we conclude by [1, Theorem 5.9] that ρ = ∂(σ) for a symbol σ in k_{2}E(t). □

Note that k_{2}E = 0 if and only if every E-quaternion algebra is split. Hence the Theorem in the introduction is covered in the following statement.

4.3. Theorem. Assume that k_{2}E ≠ 0 and that E is not real euclidean. Then we have the following:

(a) There exists ρ ∈ ℜ_{2}(E) with deg(ρ) = 4 and such that ρ ≠ ∂(σ) for any symbol σ in k_{2}E(t).

(b) There exists an E(t)-biquaternion division algebra B such that B ⊗_{E(t)} Q is not defined over E for any E(t)-quaternion algebra Q. In particular, B does not contain any E-quaternion algebra.

Proof. We claim that the inclusion

yE^{×2} ∪ (−xy)E^{×2} ⊆ xE^{×2} ∪ (x − 4)E^{×2}

cannot hold for all x, y ∈ E^{×} with {x, y} ≠ 0.

We denote by D_{E}(2) the set of nonzero sums of two squares in E and recall that for c ∈ E^{×} we have {−1, c} = 0 in k_{2}E if and only if c ∈ D_{E}(2).

If there exists a symbol σ ≠ 0 in k_{2}E that is not of the form {−1, c} for any c ∈ E^{×}, then for any x, y ∈ E^{×} with {x, y} = σ, the elements x, y, −xy represent distinct classes in E^{×}/E^{×2}, whereby the above inclusion cannot hold. If there exists c ∈ E^{×} such that c, −c ∈ E^{×} ∖ D_{E}(2), then {−1, c} ≠ 0 and for x = −1 and y = c, the above inclusion does not hold. Suppose now that {−1, −1} is the only nonzero symbol in k_{2}E. Then E^{×} = D_{E}(2) ∪ −D_{E}(2) and −1 ∉ D_{E}(2), in particular E has characteristic zero. With x = −\frac{9}{4} we obtain that xE^{×2}∪(x−4)E^{×2} = −E^{×2}. By the hypothesis, there exists some c ∈ D_{E}(2)∖E^{×2}. We set y = −c and obtain that {x, y} = {−1, −c} = {−1, −1} ≠ 0 and that the above inclusion does not hold.

Since the preliminary claim is now established, we fix elements x, y, b ∈ E^{×} with {x, y} ≠ 0 and b ∈ (yE^{×2} ∪ (−xy)E^{×2}) ∖ (xE^{×2} ∪ (x − 4)E^{×2}) . We set a = x and obtain that {a, b} = {x, y} ≠ 0 and b ∉ aE^{×2} ∪ (a − 4)E^{×2}. By Theorem 4.2 the ramification sequence

ρ = ∂({t^{2} + (a + 1)t + a, a} + {t^{2} + at + a, ab})

satisfies the claim in (a).

To show (b), we consider the E(t)-biquaternion algebra

B = (t^{2} + (a + 1)t + a, a) ⊗_{E(t)} (t^{2} + at + a, ab) .

For any f, g ∈ E(t)^{×} such that B ⊗_{E(t)} (f, g) can be defined over E, we would obtain that ρ = ∂({f, g}), in contradiction to (a). Therefore there exists no E(t)- quaternion algebra Q such that B ⊗_{E(t)} Q can be defined over E. In particular, B does not contain any E-quaternion algebra Q′, because otherwise the centraliser of Q′_{E(t)} in B would be an E(t)-quaternion algebra Q such that B ⊗_{E(t)} Q is defined over E. Since B does in particular not contain 𝕄_{2}(E), it follows that B is a division algebra. □

Acknowledgments. This work was supported by the FWO Odysseus Programme (project Explicit Methods in Quadratic Form Theory), funded by the Fonds Wetenschappelijk Onderzoek – Vlaanderen.

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Universiteit Antwerpen, Departement Wiskunde–Informatica, Middelheimlaan 1, 2020 Antwerpen, Belgium.

  E-mail address: karimjohannes.becher@uantwerpen.be