Division Algebras Over Rational Function Fields in One Variable

L.H. Rowen, A.S. Sivatski, and J.-P. Tignol

Abstract. Let A be a central simple algebra over the field of rational functions in one variable over an arbitrary field of characteristic different from 2. If the Schur index of A is not divisible by the characteristic and its ramification locus has degree at most 3, then A is Brauer-equivalent to the tensor product of a quaternion algebra and a constant central division algebra D. The index of A is computed in terms of D and the ramification of A. The result is used to construct various examples of division algebras over rational function fields.

Introduction

The Brauer group of the field of rational fractions in one variable over an arbitrary field F of characteristic zero has been described by D.K. Faddeev in [5]. To recall his result, we use the following notation: for any closed point p of the projective line ℙ_{F}^{1} , let F_{p} denote the residue field at p. Let X(F_{p}) be the character group of the absolute Galois group

2000 Mathematics Subject Classification. 16K20, 14H05. Key words and phrases. Division algebra, Brauer group, rational functions,

ramification.

Work supported in part by the European Community’s Human Potential Programme under contract HPRN–CT–2002–00287, KTAGS. Research of the first author was funded in part by an Israel Academy of Sciences Center of Excellence. The second author was supported in part by INTAS under contracts 00–566 and 99–00817. He gratefully acknowledges the hospitality of the Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, where research leading to this paper was initiated. The third author is supported in part by the National Fund for Scientific Research (Belgium).

Γ_{p} of F_{p}, i.e. the group of continuous homomorphisms

X(F_{p}) = Hom(Γ_{p}, ℚ/ℤ).

For each closed point p on ℙ_{F}^{1} , there is a map ∂_{p}: Br F (t) → X(F_{p}) known as the ramification (or residue) map at p (see [6, p. 18]). Faddeev proved that the following sequence is exact:

(0.1)  0 → Br F → Br F (t) ⟶^{⊕∂_{p}} ⨁_{p∈ℙ_{F}^{1(1)}} X(F_{p}) \underrightarrow{∑ cor} X(F) → 0,

where ∑ cor is the sum of the corestriction (or norm) maps X(F_{p}) → X(F) and ℙ_{F}^{1(1)} is the set of closed points on ℙ_{F}^{1} . See also [6, Example 9.21, p. 26], where a version in nonzero characteristic is given. If the characteristic of F is p, Faddeev’s exact sequence still holds if the Brauer groups and the character groups are replaced by their subgroups of prime-to-p torsion.

Our aim is to obtain information on the index of Brauer classes in Br F (t) from their image under the ramification map ∂ = ⊕∂_{p}. We achieve this goal in some very specific cases. To describe them, we use the following terminology:

Definition. Let

ℜ = Ker(∑ cor: ⨁_{p∈ℙ_{F}^{1(1)}} X(F_{p}) → X(F)).

The elements in ℜ are called ramification sequences. For ρ = (χ_{p}) ∈ ℜ, the support of ρ is

supp(ρ) = {p ∈ ℙ_{F}^{1(1)} | χ_{p} ≠ 0},

and χ_{p} is the component of ρ at p. Viewing supp(ρ) as a divisor on ℙ_{F}^{1} , we may consider the degree of the support of ρ:

deg supp(ρ) = ∑_{p∈supp(ρ)} deg p.

If p is a rational point (i.e., F_{p} = F), then cor: X(F_{p}) → X(F) is the identity map. Therefore, the support of a ramification sequence cannot consist of a single rational point,

deg supp(ρ) ≥ 2 for all ρ ∈ ℜ.

In this work, we consider only 2-torsion ramification sequences (and may therefore assume only char F ≠ 2). Their set is denoted by _{2}ℜ. For ρ ∈ _{2}ℜ with deg supp(ρ) = 2, we determine in Corollaries 4.2 and 4.3 the quaternion F (t)-algebras Q such that ∂(Q) = ρ. If α ∈ Br F (t) has torsion prime to char F and the same ramification sequence ρ, then by Faddev’s exact sequence (0.1) there is a central simple F -algebra A and a quaternion F (t)-algebra Q such that α is the Brauer class of A ⊗_{F} Q,

α = [A ⊗_{F} Q] ∈ Br F (t).

The index of α is determined in terms of A and ρ in Theorem 4.1.

In Section 5, we obtain similar results for ρ ∈ _{2}ℜ with deg supp(ρ) = 3. In this case, we show in Corollary 5.2 that there is up to isomorphism a unique quaternion F (t)-algebra Q_{ρ} such that

∂(Q_{ρ}) = ρ.

For any central simple F -algebra A, the index of A⊗_{F} Q_{ρ} is computed in Theorem 5.1 in terms of A and a quartic field extension E_{ρ}/F canonically associated with ρ.

The existence of quaternion F (t)-algebras with given 2-torsion ramification sequence ρ with support of degree at most 3 is also shown in a recent paper of Kunyavskiĭ, Rowen, Tikhonov and Yanchevskiĭ [9, Corollary 2.10], except in the case where the support consists of a single point of degree 3. Ramification sequences of quaternion algebras are further discussed in [9], which also gives (in Corollary 2.14) an example due to Faddeev of a ramification sequence ρ ∈ _{2}ℜ with deg supp(ρ) = 4 which is not the ramification sequence of a quaternion F (t)-algebra.

The main ingredients in the proofs of our main results are a very general observation on cochains with values in a left principal ideal domain (Section 1) and a reduction of 2-torsion ramification sequences ρ with deg supp(ρ) ≤ 3 to a normal form, which is achieved in Section 3. Although the restriction to 2-torsion elements in ℜ is not necessary for some of the reduction steps in Section 3, an example given in an appendix to this paper shows that even for 3-torsion ramification sequences the general principles of Section 1 lead to conditions which are much more difficult to handle.

1. Cochains in the ring of fractions of a left PID

This section presents the basic tool that will be used in subsequent sections to prove that certain tensor products are division algebras. As it is very general, we cast it in the setting of left Principal Ideal Domains (PID) although we apply it only in the case of polynomial rings in one indeterminate over division rings.

Let R be a left PID, with Ore ring of fractions Q,

Q = {b^{−1}a | a, b ∈ R, b ≠ 0}.

Let G be a finite group of automorphisms of R, that we extend to Q.

Lemma. Suppose (c_{σ})_{σ∈G} is a family of elements in Q^{×} such that

c_{στ}^{−1}σ(c_{τ})c_{σ} ∈ R for all σ, τ ∈ G.

There exists q ∈ Q^{×} such that

σ(q)c_{σ}q^{−1} ∈ R for all σ ∈ G.

Proof. From the definition of Q, it follows that for each σ ∈ G the left ideal

I_{σ} = {g ∈ R | σ(g)c_{σ} ∈ R}

is nonzero. As R is left Ore (as it is a left PID) and G is finite, the intersection

I = ⋂_{σ∈G} I_{σ} = {g ∈ R | σ(g)c_{σ} ∈ R for all σ ∈ G}

is a nonzero left ideal. Let q ∈ R be such that

I = Rq,

hence q ≠ 0. For all τ ∈ G we have τ(q)c_{τ} ∈ R since q ∈ I. Moreover, for σ ∈ G,

σ(τ(q)c_{τ} )c_{σ} = στ(q)σ(c_{τ})c_{σ} = (στ(q)c_{στ} )(c_{στ}^{−1}σ(c_{τ})c_{σ}).

The first factor on the right side lies in R because q ∈ I, and the second factor lies in R by hypothesis. Therefore,

σ(τ(q)c_{τ} )c_{σ} ∈ R for all σ, τ ∈ G,

hence

τ(q)c_{τ} ∈ I for all τ ∈ G,

and therefore τ(q)c_{τ} ∈ Rq for all τ ∈ G. □

The only application of this lemma in the present paper is the following: Let E be a (finite-dimensional) central division algebra over an arbitrary field K, and let x be an indeterminate over K. The ring E[x] = E ⊗_{K} K[x] is a left (and right) PID with Ore ring of fractions E(x) = E ⊗_{K} K(x). Let α be an automorphism of finite order n of E[x], and let g ∈ K[x]. Consider the algebra

(1.1)  Δ(E(x), α, g) = E(x) ⊕ E(x)y ⊕ ⋯ ⊕ E(x)y^{n−1}

where multiplication is defined by

y^{n} = g  and  yf = α(f)y for f ∈ E(x).

This algebra may be viewed as the factor ring of the skew-polynomial ring E(x)[y; α] by the ideal generated by y^{n} − g.

Theorem 1.1. Suppose n is a prime and α does not restrict to the identity on K(x). The algebra Δ(E(x), α, g) is a division algebra if and only if there is no f ∈ E[x] such that

(1.2)  α^{n−1}(f) … α(f)f = g.

Proof. By [7, Theorem 1.3.16, p. 17] (see also [1, Theorem 11.12, p. 184]), the algebra Δ(E(x), α, g) is not division if and only if there exists f ∈ E(x) for which (1.2) holds. Therefore, it suffices to show that if (1.2) holds for some f ∈ E(x), then it also holds for some f ∈ E[x].

Suppose f ∈ E(x)^{×} satisfies (1.2), and let

c_{α^{i}} = α^{i−1}(f) … α(f)f for i = 0, …, n − 1.

Then, for i, j = 0, …, n − 1,

c_{α^{i+j}}^{−1}α^{i}(c_{α^{j}})c_{α^{i}} = \begin{cases}1 & if i + j < n, \\ g & if i + j ≥ n. \end{cases}

We may therefore apply the lemma to find q ∈ E(x)^{×} such that

α(q)c_{α}q^{−1} ∈ E[x].

Let f′ = α(q)c_{α}q^{−1} = α(q)fq^{−1}. We have

α^{n−1}(f′) … α(f′)f′ = qgq^{−1} = g,

hence f′ ∈ E[x] satisfies (1.2). □

We consider two types of examples, where Δ(E(x), α, g) turns out to be the tensor product of a “constant” division algebra and a quaternion algebra. One more example is discussed in the Appendix.

1.1. Quaternion algebras with a constant slot. Let D be a central division algebra over a field F of characteristic different from 2, and let a ∈ F^{×} be such that D does not contain a square root of a. Then E = D ⊗_{F} F (\sqrt{a}) is a central division algebra over K = F (\sqrt{a}), and the nontrivial automorphism of K/F extends to an automorphism α of E which is the identity on D. Let t be an indeterminate over F and g ∈ F [t]. The algebra Δ(E(t), α, g) of (1.1) is a tensor product:

Δ(E(t), α, g) = D ⊗_{F} (a, g)_{F (t)}.

Theorem 1.1 shows that this tensor product has zero divisors if and only if there exists f ∈ E[t] such that

α(f)f = g.

Since the degree of the left side is even, the following result is clear:

Corollary 1.2. Let D be a central division F -algebra which does not contain a square root of a. For every g ∈ F [t] of odd degree, the tensor product D ⊗_{F} (a, g)_{F (t)} is a division algebra.

We now consider a case where deg g = 2.

Proposition 1.3. Let D be a central division F -algebra which does not contain a square root of a, and let b ∈ F^{×}. The tensor product D ⊗_{F} (a, t^{2} − b)_{F (t)} is not a division algebra if and only if D contains an element s such that s^{2} = ab.

Proof. As observed in the beginning of this subsection, the tensor product D ⊗_{F} (a, t^{2} − b)_{F (t)} is not a division algebra if and only if there exists f ∈ E[t] such that

(1.3)  α(f)f = t^{2} − b.

Comparing degrees, we see that if f exists it must be of the form

f = ut + v for some u, v ∈ E.

Substituting in (1.3), we obtain

(1.4)  α(u)u = 1,  α(u)v + α(v)u = 0,  α(v)v = −b.

Letting x = \sqrt{a}α(u)v, computation shows that the existence of u, v ∈ E satisfying (1.4) is equivalent to the existence of u, x ∈ E satisfying

(1.5)  α(u)u = 1,  α(x) = uxu^{−1},  x^{2} = ab.

If D contains s such that s^{2} = ab, then (1.5) holds with u = 1 and x = s.

Conversely, suppose (1.5) holds for some u, x ∈ E. If u = 1, then α(x) = x hence x ∈ D and we may set s = x. If u ≠ 1, then letting w = u − 1 ∈ E^{×} we have

α(w)u = 1 − u = −w.

For s = wxw^{−1}, it follows that

α(s) = α(w)α(x)α(w)^{−1} = α(w)uxu^{−1}α(w)^{−1} = wxw^{−1} = s,

so s ∈ D, and s^{2} = ab. □

1.2. Quaternion algebras with an indeterminate slot. Let D be a central division algebra over a field F of characteristic different from 2 and let t be an indeterminate over F. For g ∈ F [t], we may consider the tensor product D ⊗_{F} (t, g)_{F (t)} as a special case of the construction in (1.1):

D ⊗_{F} (t, g)_{F (t)} = Δ(D(x), α, g(x^{2}))

where x^{2} = t and α is the automorphism of D(x) defined by

α(x) = −x  and  α(d) = d for d ∈ D.

Therefore, Theorem 1.1 shows that this tensor product is not a division algebra if and only if there exists f ∈ D[x] such that

(1.6)  α(f)f = g(x^{2}).

We just consider one example where deg g = 3.

Proposition 1.4. Let D be a central division F -algebra and a, b, c ∈ F. The tensor product

D ⊗_{F} (t, a^{2} − bt + ct^{2} − t^{3})_{F (t)}

is not a division algebra if and only if D contains an element s such that

(1.7)  s^{4} − 2cs^{2} − 8as + (c^{2} − 4b) = 0.

Proof. Let g = a^{2} − bt + ct^{2} − t^{3} ∈ F [t]. If s ∈ D satisfies (1.7), then the polynomial

f = a + \frac{1}{2}(s^{2} − c)x + sx^{2} + x^{3}

satisfies α(f)f = g(x^{2}), hence, by the observations at the beginning of this subsection, the tensor product D⊗(t, g)_{F (t)} is not a division algebra.

Conversely, if D ⊗(t, g)_{F (t)} is not a division algebra, then (1.6) holds for some f ∈ D[x], which is necessarily of the form

f = rx^{3} + sx^{2} + ux + v for some r, s, u, v ∈ D.

Comparing the leading coefficients and the constant terms in (1.6) yields

r^{2} = 1  and  v^{2} = a^{2}.

Changing the sign of f and/or changing f into α(f) if necessary, we may assume r = 1 and v = a. Expanding (1.6) then yields the following relations between s and u:

s^{2} − 2u = c, u^{2} − 2as = b.

Eliminating u, we obtain (1.7). □

Remark. The cubic resolvent of the quartic equation (1.7) is g(t) = 0. The conceptual relation between g and the quartic polynomial in (1.7) will be made explicit in Section 5.

2. Transformations of the projective line

A choice of projective coordinates in the projective line ℙ_{F}^{1} over an arbitrary field F is an identification

ℙ_{F}^{1} = Proj(F [u, v])

where u, v are indeterminates of degree 1 over F. When projective coordinates are chosen, the rational points of ℙ_{F}^{1} are identified with F ∪ {∞} in such a way that a ∈ F corresponds to the homogeneous ideal (u − av)F [u, v], and ∞ to vF [u, v]. We then identify the field of rational functions on ℙ_{F}^{1} with F (t), where t = uv^{−1}, and ℙ_{F}^{1} \ {∞} = 𝔸_{F}^{1} = Spec(F [t]).

It is well-known that the group of transformations of ℙ_{F}^{1} is simply transitive on the triples of rational points, see for instance [2, § III.3]. In this section, we prove an analogous result for divisors consisting of a single point of degree 3, and for divisors consisting of a rational point and a point of degree 2.

For clarity, we treat separately the various cases, although a uniform proof should be possible.

Proposition 2.1. Let p, p′ be rational points on ℙ_{F}^{1} and q, q′ be closed points of degree 2. The projective transformations of ℙ_{F}^{1} which map p to p′ and q to q′ are in one-to-one correspondence with the F - isomorphisms F_{q} →^{∼} F_{q′}.

Proof. After a projective transformation, we may assume p = p′ and choose projective coordinates such that p is the point at infinity. The projective transformations which leave p invariant then are the transformations of the affine line 𝔸_{F}^{1} .

Viewing q and q′ as closed points on 𝔸_{F}^{1} = Spec(F [t]), consider monic irreducible polynomials π, π′ ∈ F [t] of degree 2 such that

q = πF [t]  and  q′ = π′F [t].

The affine transformation t ↦ at + b (with a, b ∈ F, a ≠ 0) maps q to q′ if and only if

(2.1)  π(at + b)F [t] = π′(t)F [t].

Let τ′ ∈ F_{q′} be the image of t under the canonical epimorphism F [t] → F [t]/(π′) = F_{q′}. Equation (2.1) holds if and only if π is the minimal polynomial of aτ′ + b over F. Since every element in F_{q′} has the form aτ′ + b for some a, b ∈ F, affine transformations which map q to q′ are in one-to-one correspondence with the elements in F_{q′} whose minimal polynomial is π, hence also with the F -isomorphisms F_{q} →^{∼} F_{q′}. □

Proposition 2.2. Let r, r′ be closed points of degree 3 on ℙ_{F}^{1} . The projective transformations of ℙ_{F}^{1} which map r to r′ are in one-to-one correspondence with the F -isomorphisms F_{r} →^{∼} F_{r′}.

Proof. Choose coordinates to represent ℙ_{F}^{1} as Proj(F [u, v]), where u, v are indeterminates of degree 1. We may then find homogeneous irreducible polynomials π, π′ ∈ F [u, v] of degree 3 such that

r = πF [u, v]  and  r′ = π′F [u, v].

After scaling, we may assume the coefficients of u^{3} in π and π′ are 1. The projective transformation u ↦ au + bv, v ↦ cu + dv (where a, b, c, d ∈ F and ad − bc ≠ 0) maps r to r′ if and only if

π(au + bv, cu + dv)F [u, v] = π′(u, v)F [u, v]

or, equivalently (after de-homogeneizing),

(2.2)  π(at + b, ct + d)F [t] = π′(t, 1)F [t].

Let τ′ ∈ F_{r′} be the image of t under the canonical epimorphism F [t] → F [t]/(π′(t, 1)) = F_{r′}. Equation (2.2) holds if and only if π(aτ′ + b, cτ′ + d) = 0, which means that π(t, 1) is the minimal polynomial of \frac{aτ′+b}{cτ′+d}. As in the proof of Proposition 2.1, it now suffices to establish the following result:

Claim: For every x ∈ F_{r′}, x ∉ F, there exist a, b, c, d ∈ F, uniquely determined up to a scalar factor, such that x = \frac{aτ′+b}{cτ′+d} and ad − bc ≠ 0.

Since τ′ ∉ F and x ≠ 0, we have

dim_{F} (F τ′ + F) = dim_{F} x(F τ′ + F) = 2,

hence, since dim_{F} F_{r′} = 3,

dim_{F} ((F τ′ + F) ∩ x(F τ′ + F)) ≥ 1.

If the inequality is strict, then

(2.3)  x(F τ′ + F) = F τ′ + F,

hence x ∈ F τ′ + F. Since x ∉ F, it follows that F x + F = F τ′ + F, hence τ′ ∈ F x + F and (2.3) implies that F τ′ + F is a subalgebra of F_{r′}. This is a contradiction since τ′ has degree 3. Therefore,

dim_{F} ((F τ′ + F) ∩ x(F τ′ + F)) = 1.

We may thus find a, b, c, d ∈ F, uniquely determined up to a scalar, such that aτ′ + b = x(cτ′ + d) ≠ 0, hence

x = \frac{aτ′ + b}{cτ′ + d}.

If ad − bc = 0, then x ∈ F, a contradiction. □

Viewing projective transformations as changes of projective coordinates, we readily deduce from Propositions 2.1 and 2.2 (and from the “second fundamental theorem of projective geometry” [2, § III.3]):

Corollary 2.3. (1) Let p_{1}, p_{2}, p_{3} ∈ ℙ_{F}^{1(1)} be distinct rational points and let λ_{1}, λ_{2}, λ_{3} ∈ F [u, v] be homogeneous polynomials of degree 1 which are pairwise distinct up to scalars. There is a choice of projective coordinates ℙ_{F}^{1} = Proj(F [u, v]) such that

p_{1} = λ_{1}F [u, v], p_{2} = λ_{2}F [u, v], p_{3} = λ_{3}F [u, v].

(2) Let p ∈ ℙ_{F}^{1(1)} be a rational point and q ∈ ℙ_{F}^{1(1)} be a point of degree 2. Let λ ∈ F [u, v] be a homogeneous polynomial of degree 1 and π ∈ F [u, v] be a homogeneous polynomial of degree 2 such that F_{q} ≃ F [t]/(π(t, 1)). There is a choice of projective coordinates ℙ_{F}^{1} = Proj(F [u, v]) such that

p = λF [u, v], q = πF [u, v].

(3) Let r ∈ ℙ_{F}^{1(1)} be a point of degree 3 and π ∈ F [u, v] be a homogeneous polynomial of degree 3 such that F_{r} ≃ F [t]/(π(t, 1)). There is a choice of projective coordinates ℙ_{F}^{1} = Proj(F [u, v]) such that

r = πF [u, v].

3. 2-torsion ramification sequences

In this section, we assume that the characteristic of the base field F is different from 2. We use projective transformations to set 2-torsion ramification sequences with support of degree at most 3 into a standard form.

Let Γ be the absolute Galois group of F and μ_{2} = {±1} ⊂ F^{×}. Since char F ≠ 2, we may identify the 2-torsion characters of Γ with square classes of F, via the isomorphisms

2X(F) = Hom(Γ, ℤ/2ℤ) = H^{1}(Γ, μ_{2}) = F^{×}/F^{×2}.

A ramification sequence ρ ∈ _{2}ℜ can therefore be viewed as an element in the kernel of the norm map

(3.1)  ⨁_{p∈ℙ_{F}^{1(1)}} F_{p}^{×}/F_{p}^{×2} → F^{×}/F^{×2}.

Suppose first deg supp(ρ) = 2.

Proposition 3.1. Let ρ ∈ _{2}ℜ be a 2-torsion ramification sequence with deg supp(ρ) = 2.

(1) If supp(ρ) = {p_{1}, p_{2}} (two distinct rational points), there is a nonsquare a ∈ F^{×} and a choice of projective coordinates ℙ_{F}^{1} = Proj(F [u, v]) such that

p_{1} = (u − av)F [u, v], p_{2} = (u + av)F [u, v],

and the nontrivial components of ρ are

aF^{×2} at p_{1}, aF^{×2} at p_{2}.

(2) If supp(ρ) = {q} with deg q = 2, there are nonsquares a, b ∈ F^{×} in different square classes and a choice of projective coordinates ℙ_{F}^{1} = Proj(F [u, v]) such that

q = (u^{2} − bv^{2})F [u, v]

and the component of ρ at q is aF_{q}^{×2}.

Proof. (1) The nontrivial components of ρ are square classes in F_{p_{1}} = F and F_{p_{2}} = F. These square classes coincide, since ρ is in the kernel of the norm map (3.1). We may therefore find a ∈ F^{×} such that the nontrivial components of ρ are aF^{×2} at p_{1} and p_{2}. By Corollary 2.3, we may choose projective coordinates such that p_{1} = (u−av)F [u, v] and p_{2} = (u + av)F [u, v]. This completes the proof of (1).

(2) We may find a nonsquare b ∈ F^{×} such that F_{q} ≃ F (\sqrt{b}), hence, by Corollary 2.3, there are projective coordinates such that q = (u^{2} − bv^{2})F [u, v]. The component of ρ at q is a square class in the kernel of the norm-induced map

F_{q}^{×}/F_{q}^{×2} → F^{×}/F^{×2}.

As an application of Hilbert’s Theorem 90, it is easily seen that the kernel of this map is the image of the map F^{×}/F^{×2} → F_{q}^{×}/F_{q}^{×2} induced by the inclusion F ⊂ F_{q}. (See [10, p. 202].) Therefore, there is a nonsquare a ∈ F^{×} such that the component of ρ at q is aF_{q}^{×2}. □

We now turn to the case where the support of ρ has degree 3. Three cases arise:

(I) supp(ρ) = {p_{1}, p_{2}, p_{3}} where p_{1}, p_{2}, p_{3} are three distinct ratio-

nal points; (II) supp(ρ) = {p, q} where p is a rational point and q is a point of

degree 2; (III) supp(ρ) = {r}, a single point of degree 3. In case (I), let a_{i}F^{×2} be the component of ρ at p_{i}, for i = 1, 2,

3. Since these components are nontrivial, a_{i} ∉ F^{×2} for all i. However, a_{1}a_{2}a_{3} ∈ F^{×2} since ρ is in the kernel of the norm map (3.1), hence we may substitute a_{1}a_{2} for a_{3}. Therefore, discarding the trivial components of ρ, we may write

ρ = (a_{1}F_{p_{1}}^{×2}, a_{2}F_{p_{2}}^{×2}, a_{1}a_{2}F_{p_{3}}^{×2}).

Applying Corollary 2.3, we may choose projective coordinates so that ∞ ≠ p_{1}, p_{2}, p_{3}, and in fact we may assume

p_{1} = (t − a_{1})F [t], p_{2} = (t − a_{2})F [t], p_{3} = (t − a_{1}a_{2})F [t].

In case (II), let y ∈ F_{q}^{×} be such that the component of ρ at q is yF_{q}^{×2}. Since ρ is in the kernel of the norm map (3.1), the component of ρ at p is

N_{F_{q}/F} (y)F^{×2} ∈ F^{×}/F^{×2}.

Since this component is not trivial, we have y ∉ F. Let X^{2} − mX + a be the minimal polynomial of y over F, so a = N_{F_{q}/F} (y) and

ρ = (aF_{p}^{×2}, yF_{q}^{×2}).

We may apply Corollary 2.3 to choose coordinates such that

p = (t − a)F [t], q = (t^{2} − mt + a)F [t].

In case (III), let x ∈ F_{r}^{×} be a nonsquare such that the component of ρ at r is xF_{r}^{×2},

ρ = (xF_{r}^{×2}).

Since ρ is in the kernel of the norm map (3.1), we have

N_{F_{r}/F} (x) = a^{2} for some a ∈ F^{×}.

If x ∈ F or if F_{r}/F is not separable, then a^{2} = x^{3}, and we get a contradiction since x ∉ F_{r}^{×2}. Let X^{3} − cX^{2} + bX − a^{2} be the minimal polynomial of x over F. This is an irreducible separable polynomial in F [X]. By Corollary 2.3, we may assume

r = (t^{3} − ct^{2} + bt − a^{2})F [t].

We summarize the special choice of coordinates as follows:

Proposition 3.2. In all the cases (I), (II), (III), we may choose projective coordinates in ℙ_{F}^{1} so that ∞ ∉ supp(ρ) and for each p ∈ supp(ρ) the nontrivial component of ρ at p is τF_{p}^{×2}, where τ is the image of t under the canonical map F [t] → F [t]/p.

Proof. Corollary 2.3 yields projective coordinates for which the ramification points have the required form. In case (II), the element y ∈ F_{q}^{×} is thus identified with one of the roots of X^{2}−mX+a in F [t]/q. If it is not identified with τ, Proposition 2.1 still allows a change of coordinates which leaves p and q invariant and maps y to τ. Similarly, in case (III) we may assume x ∈ F_{r}^{×} is identified with τ by Proposition 2.2. □

4. Algebras with 2-torsion ramification sequence of degree 2

In this section, the base field F is assumed to be of characteristic different from 2. Let ρ ∈ _{2}ℜ be a 2-torsion ramification sequence whose support has degree 2. Proposition 3.1 shows that, after a change of projective coordinates, we may assume that either

(I) there exists a nonsquare a ∈ F^{×} such that supp(ρ) = {a, −a}

and the nontrivial components of ρ are aF^{×2}, or (II) there exist nonsquares a, b ∈ F^{×} in different square classes such that supp(ρ) = {q = (u^{2} − bv^{2})F [u, v]} and the nontrivial component of ρ is aF_{q}^{×2}.

In each case, it is not difficult to find a quaternion F (t)-algebra Q with ramification ∂(Q) = ρ, as we proceed to show.

Recall that for any closed point p ∈ ℙ_{F}^{1(1)}, the image of a quaternion algebra (f, g)_{F (t)} under the ramification map ∂_{p} is given as follows:

(4.1)  ∂_{p}(f, g)_{F (t)} = (−1)^{v(f)v(g)}\overline{f^{v(g)}g^{−v(f)}}F_{p}^{×2} ∈ F_{p}^{×}/F_{p}^{×2},

where v is the discrete valuation of F (t) corresponding to p and denotes the residue map from the valuation ring of v to its residue field F_{p}. This follows from the description of the residue map (tame map) in Milnor’s K-theory and the functoriality of the norm residue homomorphism, see [3, Theorem 2.3]. Using this description, it is easily verified that ∂(a, t^{2}− a^{2})_{F (t)} is as in (I) above, and that ∂(a, t^{2} −b)_{F (t)} is as in (II). Therefore, if α ∈ Br F (t) has torsion prime to char F and a ramification sequence ∂(α) as in (I) (resp. (II)), then by Faddeev’s exact sequence (0.1), there exists a central simple F -algebra A such that

α = [A ⊗_{F} (a, t^{2} − a^{2})_{F (t)}] (resp. α = [A ⊗_{F} (a, t^{2} − b)_{F (t)}]).

We first compute the index ind α in both cases simultaneously by considering quaternion algebras of the type (a, t^{2} − b)_{F (t)} where a, b ∈ F^{×} satisfy a ∉ F^{×2} and ab ∉ F^{×2} (thus allowing b = a^{2}). The specific features of each case will be discussed next.

Theorem 4.1. For every central simple F -algebra A and a, b ∈ F^{×} such that a ∉ F^{×2} and ab ∉ F^{×2},

ind(A ⊗_{F} (a, t^{2} − b)_{F (t)}) = 2 gcd\{ind(A ⊗ F (\sqrt{a})), ind(A ⊗ F (\sqrt{ab}))\}.

Note that if a ∈ F^{×2} or ab ∈ F^{×2}, then (a, t^{2}−b)_{F (t)} = (ab, t^{2}−b)_{F (t)} is split, hence obviously

ind(A ⊗_{F} (a, t^{2} − b)_{F (t)}) = ind A.

Proof. By general principles (see for instance [4, p. 68]), for any quaternion F (t)-algebra Q we have

ind(A ⊗_{F} Q) divides 2 ind(A)

and, similarly, substituting A ⊗_{F} Q for A,

ind(A) = ind(A ⊗_{F} Q ⊗_{F (t)} Q) divides 2 ind(A ⊗_{F} Q).

Therefore,

(4.2)  ind(A ⊗_{F} Q) ∈ \{\frac{1}{2} ind A, ind A, 2 ind A\}.

Moreover, if ind(A ⊗_{F} Q) = \frac{1}{2} ind A, then letting D (resp. D′) be a division algebra Brauer-equivalent to A (resp. A ⊗_{F} Q), we have deg D(t) = ind A and deg D′ = \frac{1}{2} ind A, hence

deg D(t) = deg(D′ ⊗_{F (t)} Q).

Since D(t) and D′ ⊗_{F (t)} Q are Brauer-equivalent, it follows that

D(t) ≃ D′ ⊗_{F (t)} Q,

hence D(t) contains a subalgebra isomorphic to Q.

In the case where Q = (a, t^{2} − b)_{F (t)}, the last condition implies that D(t) contains elements f, g such that fg = −gf and g^{2} = t^{2} − b. Since D(t) = D ⊗_{F} F (t), we may find f_{0}, g_{0} ∈ F [t] and f_{1}, g_{1} ∈ D[t] such that f = f_{1}f_{0}^{−1} and g = g_{1}g_{0}^{−1}. Then, clearing denominators,

f_{1}g_{1} = −g_{1}f_{1}  and  g_{1}^{2} = (t^{2} − b)g_{0}^{2}.

From these equations, it follows that the leading coefficient of g_{1} is in F and that it anticommutes with the leading coefficient of f_{1}, a contradiction. Therefore, the choices in (4.2) restrict to

(4.3)  ind(A ⊗_{F} (a, t^{2} − b)_{F (t)}) ∈ {ind A, 2 ind A}.

Clearly, ind(A ⊗_{F} (a, t^{2} − b)_{F (t)}) = 2 ind A holds if and only if D ⊗_{F} (a, t^{2} − b)_{F (t)} is a division algebra, which occurs if and only if D does not contain a copy of F (\sqrt{a}) nor of F (\sqrt{ab}), by Proposition 1.3. Now, by [4, p. 67], we have

ind(A ⊗ F (\sqrt{a})) = \begin{cases}ind A & if F (\sqrt{a}) ↛ D, \\ \frac{1}{2} ind A & if F (\sqrt{a}) ↪ D. \end{cases}

Similarly,

ind(A ⊗ F (\sqrt{ab})) = \begin{cases}ind A & if F (\sqrt{ab}) ↛ D, \\ \frac{1}{2} ind A & if F (\sqrt{ab}) ↪ D. \end{cases}

Summing up, we have ind(A ⊗ (a, t^{2} − b)_{F (t)}) = 2 ind A if and only if

ind(A ⊗ F (\sqrt{a})) = ind A = ind(A ⊗ F (\sqrt{ab})).

Therefore, by (4.3), ind(A ⊗ (a, t^{2} − b)_{F (t)}) = ind A if and only if

ind(A ⊗ F (\sqrt{a})) = \frac{1}{2} ind A or ind(A ⊗ F (\sqrt{ab})) = \frac{1}{2} ind A.

□

We now consider separately the two cases for the support of the ramification sequence.

4.1. Two rational points. Suppose ρ ∈ _{2}ℜ is a ramification sequence whose support consists of two rational points. Let aF^{×2} be the component of ρ at these two points.

Corollary 4.2. The quaternion algebras Q over F (t) with ∂(Q) = ρ form a 1-parameter family. For any quaternion algebra Q in this family and any central simple F -algebra A,

ind(A ⊗_{F} Q) = 2 ind(A ⊗_{F} F (\sqrt{a})).

Proof. By Proposition 3.1, we may assume the two rational points in the support of ρ are a and −a, so that

∂(a, t^{2} − a^{2})_{F (t)} = ρ.

Now, assume Q is a quaternion F (t)-algebra such that ∂(Q) = ρ. By Faddeev’s exact sequence (0.1), there is a central simple F -algebra B such that

[Q] = [B ⊗_{F} (a, t^{2} − a^{2})_{F (t)}] in Br F (t).

Then ind(B ⊗_{F} (a, t^{2} − a^{2})_{F (t)}) = 2, and Theorem 4.1 implies that B is split by F (\sqrt{a}). Therefore, B is Brauer-equivalent to a quaternion algebra (a, λ)_{F} for some λ ∈ F^{×}, and

Q ≃ (a, λ(t^{2} − a^{2}))_{F (t)}.

For any central simple F -algebra A, the index of A ⊗_{F} Q is computed by Theorem 4.1:

ind(A ⊗_{F} Q) = ind(A ⊗_{F} (a, λ)_{F} ⊗_{F} (a, t^{2} − a^{2})_{F (t)}) =

2 ind(A ⊗_{F} (a, λ)_{F} ⊗ F (\sqrt{a})).

□

Remark. If the two rational points in the support of ρ are chosen to be 0 and ∞, the family of quaternion algebras with ∂(Q) = ρ is

{(a, λt)_{F (t)} | λ ∈ F^{×}}.

The formula ind(A ⊗_{F} (a, λt)_{F (t)}) = 2 ind(A ⊗ F (\sqrt{a})) was obtained by a different proof in [11, Proposition 2.4].

4.2. Single point of degree 2. Suppose ρ ∈ _{2}ℜ is a ramification sequence whose support consists of a single point q of degree 2.

Corollary 4.3. The quaternion F (t)-algebras Q with ∂(Q) = ρ form two 1-parameter families.

Proof. As observed in Proposition 3.1, we may find nonsquares a, b ∈ F^{×} in different square classes such that q = (u^{2} − bv^{2})F [u, v], hence F_{q} = F (\sqrt{b}), and the nontrivial component of ρ is aF_{q}^{×2}. Then,

∂(a, t^{2} − b)_{F (t)} = ρ.

If Q is a quaternion F (t)-algebra with ∂(Q) = ρ, then by Faddeev’s exact sequence (0.1) there exists a central simple F -algebra B such that

[Q] = [B ⊗_{F} (a, t^{2} − b)_{F (t)}] in Br F (t).

By Theorem 4.1, the equation ind(B⊗_{F} (a, t^{2}−b)_{F (t)}) = 2 implies that B is split by F (\sqrt{a}) or by F (\sqrt{ab}), hence B is Brauer-equivalent to (a, λ)_{F} or to (ab, λ)_{F} for some λ ∈ F^{×}. Therefore, we have either

Q ≃ (a, λ(t^{2} − b))_{F (t)}  or  Q ≃ (ab, λ(t^{2} − b))_{F (t)}.
□

Remark. Of course, the two families in the corollary above are not disjoint. For λ ∈ F^{×} there exists μ ∈ F^{×} such that

(ab, λ(t^{2} − b))_{F (t)} = (a, μ(t^{2} − b))_{F (t)}

if and only if (b, λ)_{F} is split by F (\sqrt{a}). The easy proof is omitted.

Remark. As in Corollary 4.2, we may use Theorem 4.1 to determine the index of A⊗_{F} (a, λ(t^{2}−b))_{F (t)} for any λ ∈ F^{×}, but the result depends on the parameter λ:

ind(A ⊗_{F} (a, λ(t^{2} − b))_{F (t)}) =

2 gcd\{ind(A ⊗ F (\sqrt{a})), ind(A ⊗ (a, λ)_{F} ⊗ F (\sqrt{ab}))\}.

Similarly,

ind(A ⊗_{F} (ab, λ(t^{2} − b))_{F (t)}) =

2 gcd\{ind(A ⊗ (ab, λ)_{F} ⊗ F (\sqrt{a})), ind(A ⊗ F (\sqrt{ab}))\}.

5. Algebras with 2-torsion ramification sequence of degree 3

As in the preceding section, the base field F is assumed to be of characteristic different from 2. Let ρ ∈ _{2}ℜ be a 2-torsion ramification sequence with support of degree 3. We use the same notation as in Section 3, and assume projective coordinates have been chosen as in Proposition 3.2. For each p ∈ supp(ρ), let P_{ρ} ∈ F [t] be the monic irreducible polynomial such that p = P_{p}F [t]. Let also

P_{ρ} = ∏_{p∈supp(ρ)} P_{p}, F_{ρ} = F [t]/P_{ρ}F [t] = ∏_{p∈supp(ρ)} F_{p},

and let x_{ρ} ∈ F_{ρ} be the image of t in F_{ρ}. By Proposition 3.2, we have ρ = x_{ρ}F_{ρ}^{×2}. Moreover, as observed in Section 3, F_{ρ} is an étale F -algebra.

With the notation and choice of coordinates above, it is readily verified that the quaternion algebra

(5.1)  Q = (t, −P_{ρ})_{F (t)}

satisfies ∂Q = ρ in each case (I), (II), and (III). Therefore, if α ∈ Br F (t) has torsion prime to char F and satisfies ∂α = ρ, then Faddeev’s exact sequence (0.1) yields a central simple F -algebra A such that

α = [A ⊗_{F} (t, −P_{ρ})_{F (t)}].

In order to determine the index ind α, we use a quartic étale F - algebra E_{ρ} canonically associated with ρ, viewed as the quadratic extension F_{ρ}(\sqrt{x_{ρ}})/F_{ρ}. The construction of a quartic étale F -algebra from a quadratic étale extension of a cubic étale F -algebra is discussed at length in [8, §5]. We briefly recall the cohomological version of this construction.

For any integer n, let 𝔖_{n} be the symmetric group on n elements. The conjugation action of the wreath product 𝔖_{3} ≀ 𝔖_{2} on its four Sylow 3-subgroups yields a map

s: 𝔖_{3} ≀ 𝔖_{2} → 𝔖_{4}.

Let Γ be the absolute Galois group of F. The Galois cohomology set H^{1}(Γ, 𝔖_{3}≀𝔖_{2}) (for the trivial Γ-action) classifies the isomorphism classes of quadratic extensions of cubic étale F -algebras, so ρ defines an element in this set (corresponding to the extension F_{ρ}(\sqrt{x_{ρ}})/F_{ρ}). Consider the map induced by s,

s_{∗}: H^{1}(Γ, 𝔖_{3} ≀ 𝔖_{2}) → H^{1}(Γ, 𝔖_{4}).

Since H^{1}(Γ, 𝔖_{4}) classifies quartic étale F -algebras up to isomorphism, the image of ρ under this map defines a quartic étale F -algebra E_{ρ}.

An explicit description of E_{ρ} is given in [8, §5.4]. Note that in each case (I, II or III), x_{ρ} generates F_{ρ} as an F -algebra, with minimal polynomial P_{ρ} of the form

P_{ρ} = t^{3} − ct^{2} + bt − a^{2} ∈ F [t].

(The norm N_{F_{ρ}/F} (x_{ρ}) is a square because ρ lies in the kernel of the sum of corestriction maps ⨁_{p} X(F_{p}) → X(F).) Applying [8, Corollary 5.22], we may describe E_{ρ} as the quartic F -algebra generated by an element with minimal polynomial

(5.2)  X^{4} − 2cX^{2} − 8aX + (c^{2} − 4b).

Remarkably, the classification in [8, §6.3] shows that the F -algebra E_{ρ} is a field in each case (I, II, or III). In case (I), we have

P_{ρ} = (t − a_{1})(t − a_{2})(t − a_{1}a_{2}), E_{ρ} ≃ F (\sqrt{a_{1}}, \sqrt{a_{2}}),

and (5.2) is the minimal polynomial of \sqrt{a_{1}} + \sqrt{a_{2}} + \sqrt{a_{1}a_{2}}. In case (II), E_{ρ}/F is a quartic 2-extension, with intermediate quadratic extension F (\sqrt{a}). In case (III), E_{ρ}/F is a quartic extension whose Galois closure has Galois group 𝔖_{4} or the alternating group 𝔄_{4}.

Remark. The table in [8, §6.3] is set up from the perspective of the quartic algebra. The correspondence with the notation used here is the following: Q = E_{ρ}, R(Q) = F_{ρ}, Λ_{2}(Q) = F_{ρ}(\sqrt{x_{ρ}}).

Theorem 5.1. For every central simple F -algebra A and P_{ρ} ∈ F [t] as above,

ind(A ⊗_{F} (t, −P_{ρ})_{F (t)}) = \begin{cases}ind A & if ind(A ⊗_{F} E_{ρ}) = \frac{1}{4} ind A, \\ 2 ind A & otherwise. \end{cases}

Proof. Let D be a central division F -algebra Brauer-equivalent to A. As observed in the beginning of the proof of Theorem 4.1 (see (4.2)),

ind(A ⊗_{F} (t, −P_{ρ})_{F (t)}) = \frac{1}{2} ind A, ind A or 2 ind A.

Moreover, the first case occurs only if D(t) contains a subalgebra isomorphic to (t, −P_{ρ})_{F (t)}. In particular, it must contain a square root of t, hence D(t) does not remain a division algebra over F (t)(\sqrt{t}). This is impossible, since F (t)(\sqrt{t}) is a purely transcendental extension of F. Therefore, only two possibilities remain:

ind(A ⊗_{F} (t, −P_{ρ})_{F (t)}) = ind A or 2 ind A.

The first case occurs if and only if D ⊗ (t, −P_{ρ})_{F (t)} is not a division algebra. By Proposition 1.4, this condition is equivalent to the existence of a subalgebra isomorphic to E_{ρ} in D, hence also, by [4, Theorem 12, p. 67], to

ind(D ⊗_{F} E_{ρ}) = \frac{1}{4} ind D.
□

Corollary 5.2. For any ramification sequence ρ ∈ _{2}ℜ whose support has degree 3, there is (up to isomorphism) a unique quaternion F (t)-algebra Q_{ρ} such that

∂Q_{ρ} = ρ.

Proof. By a suitable choice of projective coordinates, we may assume ρ has the special form of Proposition 3.2. As observed above (see (5.1)), the quaternion algebra

Q = (t, −P_{ρ})_{F (t)}

satisfies ∂Q = ρ. If Q′ is a quaternion F (t)-algebra such that ∂Q′ = ρ, then by Faddeev’s exact sequence (0.1), there is a central division F - algebra D such that Q′ is Brauer-equivalent to D⊗_{F} Q. By Theorem 5.1, the equality ind(D ⊗_{F} Q) = 2 leads to ind D = 1 (hence Q′ ≃ Q), or ind D = 2. The latter case occurs only if

ind(D ⊗ E_{ρ}) = \frac{1}{4} ind D = \frac{1}{2},

which is impossible. □

Thus, if ρ is in the special form of Proposition 3.2,

Q_{ρ} = (t, −P_{ρ})_{F (t)}.

The index of any Brauer class α such that ∂α = ρ is determined by Theorem 5.1.

Corollary 5.3. If a, b ∈ F^{×} are nonsquares in different square classes and D is a quaternion division F -algebra, the tensor product

D ⊗_{F} (t, a(b − t))_{F (t)}

is a division algebra.

Proof. Inspection shows that (t, a(b − t))_{F (t)} ramifies only at 0, b, and ∞, with respective ramification abF^{×2}, bF^{×2}, and aF^{×2}. Therefore, for a suitable change of variables,

(t, a(b − t))_{F (t)} = (t′, (a − t′)(b − t′)(ab − t′))_{F (t′)}.

Theorem 5.1 shows that D ⊗_{F} (t, a(b − t))_{F (t)} is not a division algebra if and only if

ind(D ⊗_{F} F (\sqrt{a}, \sqrt{b})) = \frac{1}{4} ind D = \frac{1}{2},

which is absurd. □

As a particular case, one may take F = ℚ_{p} (the field of p-adic numbers) with p ≠ 2, a ∈ ℤ_{p} a nonsquare unit, b = p and D = (a, p)_{ℚ_{p}}. Corollary 5.3 shows that the tensor product

(a, p)_{ℚ_{p}} ⊗_{ℚ_{p}} (t, a(p − t))_{ℚ_{p}(t)}

is a division algebra over ℚ_{p}(t). Note that by a theorem of Saltman [14, Theorem 3.4], division algebras of exponent 2 over ℚ_{p}(t) have index 2 or 4. The Jacob–Tignol example of a biquaternion division algebra over ℚ_{p}(t) given in the appendix of [14] is not essentially different from the example above, but the arguments used to prove it to be a division algebra are quite different.

Appendix: Tensor product with a symbol of degree 3

Theorem 1.1 has been used so far to investigate tensor products with quaternion algebras only. In this appendix, we give another application, where the prime n in the statement of Theorem 1.1 is 3 instead of 2. This will illustrate the fact that Theorem 1.1 leads to more complex conditions for n > 2.

Let F be a field of characteristic different from 3, containing a primitive cube root of unity ζ. For a, b ∈ F^{×}, we denote by (a, b)_{ζ,F} the symbol algebra of degree 3 generated by two elements i, j subject to

i^{3} = a,  ji = ζij,  j^{3} = b.

(See [4, §11], where this type of algebra is called power norm residue algebra.)

Proposition 5.4. Let D be a central division F -algebra, let a, b ∈ F^{×} and let t be an indeterminate over F. The tensor product

(5.3)  D ⊗_{F} (t, at + b)_{ζ,F (t)}

is not a division algebra if and only if D contains elements u, v satisfying

\begin{matrix}u^{3} = a, & u(uv − vu) = ζ(uv − vu)u, \\ v^{3} = b, & v(uv − vu) = ζ^{2}(uv − vu)v. \end{matrix} ; (5.4)

Proof. The tensor product (5.3) can be viewed as a special case of the Δ-construction in (1.1),

D ⊗_{F} (t, at + b)_{ζ,F (t)} = Δ(D(x), α, ax^{3} + b)

where x^{3} = t and α is the automorphism of D(x) defined by

α(x) = ζx, α(d) = d for d ∈ D.

(Compare Section 1.2.) By Theorem 1.1, this algebra is not a division algebra if and only if there exists f ∈ D[x] such that

(5.5)  α^{2}(f)α(f)f = ax^{3} + b.

Comparing degrees, it is clear that f must be of the form f = ux + v for some u, v ∈ D if it exists. Comparing coefficients of like powers of x in (5.5) (and using ζ^{2} = −1 − ζ) yields (5.4). □

For instance, the tensor product (5.3) is not a division algebra if D ⊃ F (\sqrt[3]{a}, \sqrt[3]{b}), since then (5.4) has a solution where u and v commute. This is not the only case, however:

Corollary 5.5. The tensor product

(b, a)_{ζ,F} ⊗_{F} (t, at + b)_{ζ,F (t)}

is not a division algebra.

Proof. The elements u, v ∈ (b, a)_{ζ,F} such that

v^{3} = b,  uv = ζvu,  u^{3} = a

satisfy (5.4). □

A sharp contrast is given by the following positive result:

Corollary 5.6. Suppose a, b are indeterminates over a field k containing a primitive cube root of unity, and let F = k(a, b). The tensor product

(a, b)_{ζ,F} ⊗_{F} (t, at + b)_{ζ,F (t)}

is a division algebra.

Proof. Embedding F into the field of iterated Laurent series F̂ = k((a))((b)), we show the stronger result that

(a, b)_{ζ,F̂} ⊗F̂ (t, at + b)_{ζ,F̂(t)}

is a division algebra.

Observe that the canonical (a, b)-adic valuation on F̂ (with values in ℤ × ℤ) extends to a totally ramified valuation ν on (a, b)_{ζ,F̂}. The standard generators i, j of (a, b)_{ζ,F̂} satisfy

ν(i) = (\frac{1}{3}, 0), ν(j) = (0, \frac{1}{3}).

Suppose u, v ∈ (a, b)_{ζ,F̂} satisfy (5.4). Then

ν(u) = (\frac{1}{3}, 0) = ν(i), ν(v) = (0, \frac{1}{3}) = ν(j),

hence, by [12, §3], letting denote the residue map,

\overline{v^{−1}u^{−1}vu} = \overline{j^{−1}i^{−1}ji} = ζ

and

\overline{v^{−1}u^{−2}vu^{2}} = \overline{j^{−1}i^{−2}ji^{2}} = ζ^{2}.

On the other hand, from the equation

u(uv − vu) = ζ(uv − vu)u

it follows that

u^{2}v + ζ^{2}uvu + ζvu^{2} = 0

hence, dividing by u^{2}v,

1 + ζ^{2}v^{−1}u^{−1}vu + ζv^{−1}u^{−2}vu^{2} = 0.

Taking the residue of each side yields 3 = 0. □

Remark. Letting K = k(a, b, t) = F (t) and

D_{1} = (a, b)_{ζ,K}, D_{2} = (t, at + b)_{ζ,K},

Corollaries 5.5 and 5.6 show that D_{1} ⊗_{K} D_{2} is a division algebra while D_{1}^{op} ⊗_{K} D_{2} is not a division algebra. The first examples of this type are in [12, §5]. The example above is essentially the same as the Tignol– Wadsworth example in degree 3.

Remark. Suppose D is a central division F -algebra of degree 3. If u, v ∈ D satisfy (5.4) and uv − vu ≠ 0, then

(v, uv − vu, u)

is a chain of length 2 of Kummer elements, in the sense of Rost [13]. The example in [13, Appendix] of a division algebra containing two Kummer elements which cannot be related by a chain of length 2 yields, by Proposition 5.4, a division algebra of exponent 3 and degree 9 over the field of rational fractions in two indeterminates 𝔽_{7}(t_{1}, t_{2}).

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Bar-Ilan University, Ramat Gan, 52100 Israel

E-mail address: rowen@macs.biu.ac.il

St Petersburg Electrotechnical University, 197376 St Petersburg, Russia

E-mail address: sivatsky@as3476.spb.edu

Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

E-mail address: tignol@math.ucl.ac.be
URL: http://www.math.ucl.ac.be/tignol