ALGEBRAS OF ODD DEGREE WITH INVOLUTION, TRACE FORMS AND DIHEDRAL EXTENSIONS^{1}

Darrell Haile, Max-Albert Knus, Markus Rost and Jean-Pierre Tignol^{2}

Dedicated to the memory of S.A. Amitsur

Abstract A 3-fold Pfister form is associated to every involution of the second kind on a central simple algebra of degree 3. This quadratic form is associated to the restriction of the reduced trace quadratic form to the space of symmetric elements; it is shown to classify involutions up to conjugation. Subfields with dihedral Galois group in central simple algebras of arbitrary odd degree with involution of the second kind are investigated. A complete set of cohomological invariants for algebras of degree 3 with involution of the second kind is given.

1. Introduction. Let B be a central simple algebra over a field K with an involution σ of the second kind and let F be the fixed subfield of K. Let Trd be the reduced trace and Nrd the reduced norm of B. The restriction Q_{σ} of the trace form Q: (x, y) ↦ Trd(xy) to the F –space (B, σ)_{+} of symmetric elements of B is a quadratic form with values in F. It is an invariant of σ and the aim of this paper is to study this invariant. We first describe the general form of Q_{σ} for algebras of arbitrary odd degree and then restrict our attention to central simple algebras of degree 3.

Consider a cubic étale F -subalgebra L ⊂ (B, σ)_{+}. The restriction of the trace form to L is nonsingular, hence we have an orthogonal decomposition:

(B, σ)_{+} = L ⊥ V.

1The text appeared as: Israel J. Math. 96 B, 299-340 (1996) - Amitsur Volume

2Supported in part by the National Fund for Scientific Research (Belgium)

We give an explicit description of the restriction of the form Q_{σ} to V, using a special case of a construction introduced by T. Springer [19], in connection with exceptional Jordan algebras, and generalized by Petersson–Racine [11] to algebras of degree 3. Our next goal is to show that the trace form Q_{σ} determines the involution up to isomorphism. As a consequence we get a parametrization of all involutions of the second kind on a central simple algebra of degree 3 which leave elementwise invariant a given cubic separable subalgebra. We also associate a 3-fold Pfister form π(B, σ) to Q_{σ}, which determines σ up to isomorphism, and characterize the class of involutions for which this 3-fold Pfister form is hyperbolic. The existence of such involutions, which we call distinguished, follows from Springer’s construction, but is also related to a crossed product construction given by A.A. Albert [2]. A distinguished involution is characterized by the fact that the space (B, σ)_{+} contains up to isomorphism every cubic étale F -subalgebra of B.

In the last section, we use Galois cohomology and symbols to analyze étale subalgebras of dihedral central simple algebras. As applications, we get on one hand different proofs of previous results of the paper and on the other hand we show that a dihedral algebra of degree 2n, n odd, is cyclic if a quadratic extension of F contains a primitive n^{th}–root of 1 (see Corollary 30 for the precise statement). This is due to L. Rowen and D. Saltman [15] if F contains a primitive n^{th}–root of 1. As a last application, we give a complete set of cohomological invariants for algebras of degree three with involution of the second kind.

The second author is indebted to H.P. Petersson for useful comments on the subject.

2. Some General Results. Throughout the paper, B denotes a central simple algebra over a field K of characteristic different from 2 and σ denotes an involution of the second kind on B, i.e. a map σ: B → B such that

σ(x + y) = σ(x) + σ(y),  σ(xy) = σ(y)σ(x),  σ^{2}(x) = x

for all x, y ∈ B, and σ|_{K} ≠ I_{K}. We let F denote the subfield of K elementwise invariant under σ and denote by : K → K the restriction of σ to K.

Under a scalar extension of F, the field K — hence also the algebra B — may split into a direct product of two factors. Therefore, we shall also allow K to be a split quadratic étale F -algebra:

K = F × F.

In that case, B = A × A′ for some central simple F -algebras A, A′ which are exchanged under the involution σ. Therefore, there is an isomorphism of K-algebras with involution (i.e. an isomorphism which commutes with the involutions):

(B, σ) ≃ (A × A^{op}, s)

where A^{op} is the opposite algebra of A and s is the switch involution:

s(a_{1}, a_{2}^{op}) = (a_{2}, a_{1}^{op}).

Abusing the terminology, we shall also consider (B, σ) as a central simple algebra with involution in this case. (It is indeed simple as an algebra-with-involution: see Jacobson’s definition in [8, p. 208]).

We let α ∈ F^{×} be such that K = F (\sqrt{α}) = F [X]/(X^{2} − α). In particular, we have α ∈ F^{×}^{2} if K = F × F.

Let (B, σ)_{+} denote the F -vector space of σ-symmetric elements:

(B, σ)_{+} = {b ∈ B: σ(b) = b}.

We denote by Q_{σ} the restriction to (B, σ)_{+} of the reduced trace quadratic form:

Q_{σ}(x) = Trd_{B}(x^{2}) for x ∈ (B, σ)_{+}.

For any u ∈ (B, σ)_{+} ∩B^{×}, σ′ = Int(u)∘σ is again an involution of the second kind of B and conversely, if σ, σ′ are involutions of the second kind of B, there exists u ∈ (B, σ)_{+} ∩ B^{×} such that σ′ = Int(u) ∘ σ. Let ⟨u⟩_{B} be the B-hermitian form on B (as a right B–module) given by

⟨u⟩_{B}(x, y) = σ(x)uy,

for u ∈ (B, σ)_{+} ∩ B^{×} and x, y ∈ B. A right B-module automorphism

r_{v}: x ↦ vx, v ∈ B^{×}, of B is an isometry ⟨u_{1}⟩_{B} →^{∼} ⟨u_{2}⟩_{B} if σ(v)u_{2}v =

u_{1} and is a similarity if there is λ ∈ F^{×} such that λσ(v)u_{2}v = u_{1}.

Lemma 1. Let u_{1}, u_{2} ∈ (B, σ)_{+} ∩ B^{×} and let σ_{i} = Int(u_{i}) ∘ σ. Then

(1) An isomorphism (B, σ_{1}) →^{∼} (B, σ_{2}) induces an isometry Q_{σ_{1}} →^{∼} Q_{σ_{2}}.

(2) (B, σ_{1}) and (B, σ_{2}) are isomorphic (as K-algebras with involution) if and only if the hermitian spaces ⟨u_{1}⟩_{B} and ⟨u_{2}⟩_{B} are similar.

Proof. (1): If Int(v): (B, σ_{1}) →^{∼} (B, σ_{2}) is an isomorphism, then Int(v)[(B, σ_{1})_{+}] = (B, σ_{2})_{+} and

Q_{σ_{2}}(vxv^{−1}, vyv^{−1}) = Trd_{B}(vxyv^{−1}) = Q_{σ_{1}}(x, y)

for x, y ∈ (B, σ)_{+}. (2): The automorphism Int(v) of B is an isomorphism (B, σ_{1}) →^{∼} (B, σ_{2}), if and only if σ_{2}(vxv^{−1}) = vσ_{1}(x)v^{−1} for all x ∈ B, if and only if u_{2} = λvu_{1}σ(v) for some λ ∈ F^{×}, hence ⟨u_{1}⟩_{B} and ⟨u_{2}⟩_{B} are similar. Conversely any such similitude induces an isomorphism

(B, σ_{1}) →^{∼} (B, σ_{2}).

Let B = M_{n}(K) be split and let τ(x_{ij}) = (\overline{x}_{ij})^{t}, where t is transpose and x ↦ \overline{x} is conjugation on K. Any u ∈ (M_{n}(K), τ)_{+} is a hermitian matrix, hence there is v ∈ GL_{n}(K) such that σ(v)uv = a = diag(α_{1}, …, α_{n}), α_{i} ∈ F^{×}. Thus any involution of M_{n}(K) of the second kind is isomorphic to an involution of the form σ = Int(a) ∘ τ with a = diag(α_{1}, ⋯, α_{n}), α_{i} ∈ F^{×}.

Let h_{a} = ⟨α_{1}, …, α_{n}⟩_{K} be the hermitian form on K^{n} determined by diag(α_{1}, ⋯, α_{n}), i.e. h_{a}(x, y) = ∑ \overline{x}_{i}α_{i}y_{i} = \overline{x}^{t}ay, x, y ∈ K^{n}. Any isometry h_{a} →^{∼} h_{a′} of the K–space K^{n} can be viewed as an isometry

⟨a⟩_{M_{n}(K)} →^{∼} ⟨a′⟩_{M_{n}(K)} of the M_{n}(K)–space M_{n}(K).

For α_{1}, …, α_{n} ∈ F^{×}, we denote by ⟨α_{1}, …, α_{n}⟩ the quadratic form on F^{n} determined by diag(α_{1}, ⋯, α_{n}), and by ⟨⟨α_{1}, …, α_{n}⟩⟩ the nfold Pfister form:

⟨⟨α_{1}, …, α_{n}⟩⟩ = ⟨1, −α_{1}⟩ · · · · · ⟨1, −α_{n}⟩.

The next proposition follows by straightforward computation.

Proposition 2. Let K = F (\sqrt{α}). For a = diag(α_{1}, ⋯, α_{n}) ∈ M_{n}(K) and σ = Int(a) ∘ τ we have

Q_{σ} ≃ n⟨1⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · (⊥_{1≤i<j≤n} ⟨α_{i}α_{j}⟩).

In order to get a similar result for arbitrary central simple algebras of odd degree, we first prove:

Lemma 3. Let L/F be a field extension of odd degree and let q be a quadratic form over F. Let also q_{L} denote the quadratic form over L derived from q by extending scalars to L, and let α ∈ F^{×} ∖ F^{×}^{2}. If q_{L} ≃ ⟨⟨α⟩⟩ · h for some quadratic form h over L, of determinant 1, then there is a quadratic form t of determinant 1 over F such that

q ≃ ⟨⟨α⟩⟩ · t.

Proof. Let K = F (\sqrt{α}) and E = L · K = L(\sqrt{α}). Let also q_{an} denote an anisotropic form over F which is Witt-equivalent to q. The form (q_{an})_{E} is Witt-equivalent to the form (⟨⟨α⟩⟩·h)_{E}, hence it is hyperbolic. Since the field extension E/K has odd degree, Springer’s theorem on the behaviour of quadratic forms under field extensions of odd degree [16, Theorem 2.5.3] shows that (q_{an})_{K} is hyperbolic, hence, by [16, Remark 2.5.11],

q_{an} = ⟨⟨α⟩⟩ · t_{0}

for some quadratic form t_{0} over F. Let dim q = 2d, so that dim h = d, and let w denote the Witt index of q, so that

q ≃ wℍ ⊥ ⟨⟨α⟩⟩ · t_{0},  (1)

where ℍ is the hyperbolic plane. We then have dim t_{0} = d − w, hence

det q = (−1)^{w} (−α)^{d−w} · F^{×}^{2} ∈ F^{×}/F^{×}^{2}.

On the other hand, the relation q_{L} ≃ ⟨⟨α⟩⟩ · h yields:

det q_{L} = (−α)^{d} · L^{×}^{2} ∈ L^{×}/L^{×}^{2}.

Therefore, α^{w} ∈ F^{×} becomes a square in L; since the degree of L/F is odd, this implies that α^{w} ∈ F^{×}^{2}, hence w is even. Letting t_{1} = \frac{w}{2} ℍ ⊥ t_{0}, we then derive from (1):

q ≃ ⟨⟨α⟩⟩ · t_{1}.

It remains to prove that we may modify t_{1} so as to satisfy the determinant condition. Since dim t_{1} = d, we have modulo the square I^{2}F of the fundamental ideal of the Witt ring of F:

t_{1} ≡ \{ \begin{matrix}⟨⟨(−1)^{d(d−1)/2} det t_{1}⟩⟩ & if d is even \\ ⟨(−1)^{d(d−1)/2} det t_{1}⟩ & if d is odd. \end{matrix}

We may use these relations to compute the Clifford algebra of q ≃ ⟨⟨α⟩⟩ · t_{1} (up to Brauer-equivalence): in both cases we get the same quaternion algebra:

C(q) ∼ (α, (−1)^{d(d−1)/2} det t_{1})_{F} .

On the other hand, since det h = 1 we derive from q_{L} ≃ ⟨⟨α⟩⟩ · h:

C(q_{L}) ∼ (α, (−1)^{d(d−1)/2})_{L}.

Therefore, the quaternion algebra (α, det t_{1})_{F} is split, since it splits over the extension L/F of odd degree. Therefore, if δ ∈ F^{×} is a representative of det t_{1} ∈ F^{×}/F^{×2}, we have

δ ∈ n_{K/F} (K^{×}).

Let β ∈ F^{×} be a represented value of t_{1}, so that

t_{1} ≃ t_{2} ⊥ ⟨β⟩

for some quadratic form t_{2} over F, and let

t = t_{2} ⊥ ⟨δβ⟩.

Then

det t = δ · det t_{1} = 1.

On the other hand, since δ is a norm from the extension K/F we have

⟨⟨α⟩⟩ · ⟨δβ⟩ ≃ ⟨⟨α⟩⟩ · ⟨β⟩,

hence

⟨⟨α⟩⟩ · t ≃ ⟨⟨α⟩⟩ · t_{1} ≃ q.

Proposition 4. Let B be a central simple K-algebra of odd degree n = 2m − 1 with an involution σ of the second kind. There is a quadratic form q_{σ} of dimension n(n − 1)/2 and determinant 1 over F such that

Q_{σ} ≃ n ⟨1⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · q_{σ}.

Proof. Suppose first K = F × F. We may then assume (B, σ) = (A × A^{op}, s), where s is the switch involution. In that case

(B, σ)_{+} = {(a, a^{op}): a ∈ A} ≃ A,

and Q_{σ} is isometric to the reduced trace quadratic form on A. Since α ∈ F^{×}^{2}, we have to show that this quadratic form is Witt-equivalent to n⟨1⟩. By Springer’s theorem, it suffices to prove this relation over an odd-degree field extension. Since the degree of A is odd, we may therefore assume A is split: A = M_{n}(F). In that case, the relation is easy to check. (Observe that the upper-triangular matrices with zero diagonal form a totally isotropic subspace).

For the rest of the proof, we may thus assume K is a field. Let D be a division K-algebra Brauer-equivalent to B and let τ: D → D be an involution of the second kind on D. Let also L be a field contained in (D, τ)_{+} and maximal for this property. The field E = L·K is then a maximal subfield of D, otherwise the centralizer C_{D}E contains a symmetric element outside E, contradicting the maximality of L. We have [L : F] = [E : K] = deg D, hence the degree of L/F is odd, since D is Brauer-equivalent to the algebra B of odd degree. Moreover, the algebra

B ⊗_{F} L = B ⊗_{K} E

splits, since E is a maximal subfield of D. By Proposition 2 the quadratic form [Q_{σ}]_{L} obtained from Q_{σ} by scalar extension to L has the form

[Q_{σ}]_{L} ≃ n ⟨1⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · h  (2)

where h =⊥_{1≤i<j≤n} ⟨α_{i}α_{j}⟩ for some α_{1}, …, α_{n} ∈ L^{×}. Therefore, the Witt index of the form [Q_{σ}]_{L} ⊥ n ⟨−1⟩ is at least n:

w([Q_{σ}]_{L} ⊥ n ⟨−1⟩) ≥ n.

By Springer’s theorem the Witt index of a form does not change under an odd-degree scalar extension. Therefore,

w(Q_{σ} ⊥ n ⟨−1⟩) ≥ n,

and it follows that Q_{σ} contains a subform isometric to n ⟨1⟩. Let

Q_{σ} ≃ n ⟨1⟩ ⊥ q

for some quadratic form q over F. Relation (2) shows that

(q)_{L} ≃ ⟨2⟩ · ⟨⟨α⟩⟩ · h.

Since det h = 1, we may apply Lemma 3 to the quadratic form ⟨2⟩ · q and get a quadratic form q_{σ} over F, of determinant 1, such that

⟨2⟩ · q ≃ ⟨⟨α⟩⟩ · q_{σ},

hence

Q_{σ} ≃ n⟨1⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · q_{σ}.

We conclude with a result which will be used in the next section for algebras of degree 3.

Lemma 5. Let L ⊂ (B, σ)_{+} be étale of dimension n over F and let R = L ⊗_{F} L ⊗_{F} K. Then B is a free R-module of rank one via left and right multiplication; the action is equivariant with respect to the involution σ on B and the action σ: R → R given by σ(λ ⊗ μ ⊗ x) = μ ⊗ λ ⊗ x̄. In particular we have an induced action of R^{σ} on (B, σ)_{+} Proof. We may assume that F is separably closed, that

B = M_{n}(F) × M_{n}(F)^{op},

σ is the switch involution, and

L = {(d, d^{op}): d is diagonal}.

Let L′ be the set of diagonal matrices in M_{n}(F). We have

M_{n}(F) = L′ ⊕ xL′ ⊕ ⋯ ⊕ x^{n−1}L′,

with x a permutation matrix of order n. In this case ξ = 1 + x ⋯ + x^{n−1} is a free generator of M_{n}(F) as a L′ ⊗ L′-module. Thus (ξ, ξ^{op}) is a free generator of B as L ⊗ L-module. The last claim follows by a direct verification.

3. A Construction of Springer. In this section, we restrict attention to central simple algebras of degree 3 over a field of characteristic different from 2, 3. We use the same notation as in the preceding section; in particular, we denote by B a central simple K-algebra with involution σ of the second kind. Consider a cubic étale F -subalgebra L ⊂ (B, σ)_{+} and denote by t_{L/F}: L → F and n_{L/F}: L → F the trace, resp. the norm of L. The restriction of the trace form to L is nonsingular, hence we have an orthogonal decomposition:

(B, σ)_{+} = L ⊥ V.

For v ∈ V let

N(v) = \frac{1}{2} Trd_{B}(v^{2}) − p_{L}(v^{2}) ∈ L,

where p_{L}: (B, σ)_{+} → L denotes the orthogonal projection. We define an L-action on V such that (V, N) is a nonsingular quadratic space of rank 2 over L. We give two descriptions of the action of L on V. The first was introduced by T. Springer [19] in connection with exceptional Jordan algebras and generalized by Petersson–Racine [11] to Jordan algebras of degree 3. The second uses Lemma 5 and is specific to central simple algebras of degree 3.

Lemma 6. (Springer’s Construction) The space V is a free Lmodule of rank 2 through the operation

(ℓ, v) ↦ ℓ ∘ v = t_{L/F} (ℓ)v − ℓv − vℓ ∈ V

and N: V → L is a nonsingular quadratic form for this structure. Moreover, for all v ∈ V,

Q_{σ}(v) = 2 t_{L/F} (N(v)).

If v ∈ V is invertible in B, then

Trd_{B}(v^{−1}ℓ) = − Nrd_{B}(v)^{−1}t_{L/F} (N(v)ℓ)

for all ℓ ∈ L.

Proof. Extending scalars from F to an algebraic closure, we may assume that B is of the form M_{3}(F) × M_{3}(F)^{op}, σ is the switch involution, and L = {(d, d^{op}): d is diagonal}. In this case the lemma follows by explicit computation.

We now describe the second construction: Let D = F (\sqrt{δ}) be the discriminant algebra of L, i.e. δ is the determinant of the trace form t_{L/F} (x^{2}) = Q_{σ}|_{L}(x). There exists a decomposition

L ⊗_{F} L = L × L ⊗_{F} D,  (3)

such that the twist σ of L ⊗_{F} L restricts on D to a ↦ ā. Note that there are three natural imbeddings L → L ⊗_{F} D. Two of them are given by λ ↦ pr(λ ⊗ 1) and λ ↦ pr(1 ⊗ λ), respectively (pr is the projection L ⊗_{F} L → L ⊗_{F} D) and L → L ⊗_{F} D, λ ↦ λ ⊗ 1, is the third one, which is σ-invariant. We have an induced decomposition

R = L ⊗_{F} L ⊗_{F} K = L ⊗_{F} K × L ⊗_{F} D ⊗_{F} K.  (4)

The σ-action on R (see Lemma 5) restricts on L ⊗_{F} D ⊗_{F} K to λ ⊗ d ⊗ a ↦ λ ⊗ d̄⊗ ā, so the fixed subalgebra is

R^{σ} = L × L ⊗_{F} H,

where H = F (\sqrt{αδ}).

Lemma 7. The decomposition of the R-module B induced by the decomposition (4) reduces to the decomposition

B = L ⊗_{F} K ⊥ V ⊗_{F} K

over K. In particular R^{σ} = L × L ⊗_{F} H acts on (B, σ)_{+} = L ⊥ V componentwise and V is a free L ⊗ H-module of rank one. Moreover the action given by the restriction L ⊂ L ⊗ H coincide with the Springer action, hence N: V → L is a nonsingular quadratic form on the free L-module V of rank 2. We have N(hv) = n_{H/F} (h)N(v) for h ∈ H, so that N extends to a hermitian form on the L ⊗_{F} Hspace V of rank one.

Proof. As for Lemma 6 it suffices to check the split case, where the claims follow by explicit computations.

Let, as above, δ denote the discriminant of L. We recall that δ ∈ F^{×}^{2} if and only if L is cyclic and that

Q_{σ}|_{L} = ⟨1, 2, 2δ⟩;

this is easy to check if L is not a field, since then L = F ×F (\sqrt{δ}). The general case follows by extending scalars from F to L and applying Springer’s theorem. Combining this result with Lemma 6, we get

Q_{σ} = ⟨1, 2, 2δ⟩ ⊥ ⟨2⟩ · t_{L/F} (N),  (5)

where t_{L/F} (N) denotes the Scharlau transfer of N.

Proposition 8. Let K = F (\sqrt{α}). The following conditions are equivalent:

(a) N is hyperbolic.

(b) δ = α in F^{×}/F^{×}^{2}.

(c) LK is a cyclic extension of K and is Galois with Galois group S_{3} over F.

Moreover, if these conditions hold, then B is a crossed product:

B = LK ⊕ LKx ⊕ LKx^{2}

for some x such that x^{3} ∈ F and σ(x) = x.

Proof. (a) ⇔ (b): follows immediately from Lemma 7. Alternatively, if we only want to use Springer’s Construction (Lemma 6), we get (a) ⇒ (b) as follows: If N is hyperbolic, equation (5) yields:

det Q_{σ} = −δ.

On the other hand, Proposition 4 shows that det Q_{σ} = −α. Therefore, (b) follows.

(b) ⇒ (c): If δ = α, then the discriminant of L becomes a square in K, hence LK/K is cyclic. Let ρ denote a generator of the Galois group Gal(LK/K). The restriction of σ to LK is an automorphism of order 2 of LK over F, hence σ|_{LK} and ρ generate a group of automorphisms of order at least 6 of LK/K which shows that LK/F is Galois.

If α ≠ 1, then L/F is not cyclic, hence the group generated by ρ and σ|_{LK} is not cyclic. It is therefore isomorphic to S_{3}.

If α = δ = 1, then B ≃ A × A^{op} and σ is isomorphic to the switch involution. Moreover, LK ≃ L_{0}×L_{0}^{op} for some cyclic extension L_{0}/F. If γ is a generator of the Galois group Gal(L_{0}/F), then we choose for ρ the automorphism of L given by:

ρ(ℓ_{1}, ℓ_{2}^{op}) = (γ(ℓ_{1}), γ^{2}(ℓ_{2})^{op}).

Thus, σ and ρ do not commute; they generate a group isomorphic to S_{3}.

Our next goal is to show that B contains an invertible element x such that σ(x) = x and xℓ = ρ(ℓ)x for all ℓ ∈ LK. These relations imply that x^{3} centralizes LK, is σ-symmetric and commutes with x, hence x^{3} ∈ F. Let

S = {x ∈ (B, σ)_{+}: xℓ = ρ(ℓ)x for all ℓ ∈ LK};

this is a vector space over F in which the invertible elements form a Zariski open set. In order to prove that this set is non-empty, we may extend scalars from F to an algebraic closure, and assume

B = M_{3}(F) × M_{3}(F)^{op},

σ is the switch involution, and

L = {(d, d^{op}): d is diagonal}.

We may further assume that ρ maps

(( \begin{matrix}d_{1} \\ & d_{2} \\ & & d_{3} \end{matrix} ) , ( \begin{matrix}d′_{1} \\ & d′_{2} \\ & & d′_{3} \end{matrix} )^{op})

to

(( \begin{matrix}d_{3} \\ & d_{1} \\ & & d_{2} \end{matrix} ) , ( \begin{matrix}d′_{2} \\ & d′_{3} \\ & & d′_{1} \end{matrix} )^{op});

we may then choose

x = (( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} )^{op}) ∈ S ∩ B^{×},

proving the claim. It follows that B is a crossed product, as required.

(c) ⇒ (a): For every element x as above, we have x, x^{2} ∈ V (as is easily seen by scalar extension to an algebraic closure of F), hence N(x) = 0, proving that N is isotropic, hence hyperbolic since its dimension over L is 2.

Remark. Proposition 8 can also be deduced from the Corollary of [12, Theorem 1] in relation with [11, Proposition 2.2].

In order to give an explicit description of the form N for general L, we need the following lemmas:

Lemma 9. Let p_{V}: (B, σ)_{+} → V denote the orthogonal projection. For all v ∈ V,

n_{L/F} (N(p_{V} (v^{2}))) = n_{L/F} (N(v))^{2}.

Proof. The lemma follows by explicit computation after extending scalars from F to an algebraic closure.

Lemma 10. For any λ ∈ L^{×} such that n_{L/F} (λ) ∈ F^{×}^{2}, the quadratic form

⟨⟨δ⟩⟩ · [t_{L/F} (⟨λ⟩) ⊥ ⟨−1⟩]

is hyperbolic.

Proof. By Springer’s theorem, it suffices to prove that the quadratic form above is hyperbolic after extending scalars from F to L. We may thus assume L = F × F′ where F′ = F (\sqrt{δ}). Let λ = (λ_{0}, λ_{1}) with λ_{0} ∈ F and λ_{1} ∈ F′; then

t_{L/F} (⟨λ⟩) = ⟨λ_{0}⟩ ⊥ t_{F′/F} (⟨λ_{1}⟩).  (6)

By [16, p. 50], the image of the transfer map from the Witt ring W F′ to W F is killed by ⟨⟨δ⟩⟩, hence

⟨⟨δ⟩⟩ · t_{F′/F} (⟨λ_{1}⟩) = 0 in W F.  (7)

On the other hand,

n_{L/F} (λ) = λ_{0} n_{F′/F} (λ_{1}) ∈ F^{×}^{2},

hence λ_{0} is a norm from F′ to F, and therefore

⟨⟨δ⟩⟩ · ⟨λ_{0}, −1⟩ = 0 in W F.  (8)

The lemma follows from (7) and (8), in view of the decomposition (6).

Proposition 11. The quadratic form N has a diagonalization

⟨⟨αδ⟩⟩ · ⟨λ⟩

for some λ ∈ L^{×} such that n_{L/F} (λ) ∈ F^{×}^{2}. Consequently,

Q_{σ} = ⟨1, 2, 2δ⟩ ⊥ ⟨2⟩ · ⟨⟨αδ⟩⟩ · t_{L/F} (⟨λ⟩)
= ⟨1, 1, 1⟩ ⊥ ⟨2δ⟩ · ⟨⟨α⟩⟩ · t_{L/F} (⟨λ⟩)

for some λ ∈ L^{×} such that n_{L/F} (λ) ∈ F^{×}^{2}.

Proof. Let v ∈ V be such that N(v) is invertible in L. Lemma 9 shows that N(p_{V} (v^{2})) also is invertible. The element p_{V} (v^{2}) is a basis of V as a L ⊗ H-module, so that by Lemma 7, letting λ = N(p_{V} (v^{2})), we get

N = ⟨⟨αδ⟩⟩ · ⟨λ⟩

as wanted. Alternatively, without using Lemma 7, and letting λ = N(p_{V} (v^{2})) as above, we have

N = ⟨λ, λ′⟩

for some λ′ ∈ L^{×}, and n_{L/F} (λ) = n_{L/F} (N(v))^{2} ∈ F^{×}^{2}, by Lemma 9. On the other hand, Proposition 8 shows that N becomes hyperbolic over L(\sqrt{αδ}), so det N = −αδ, and therefore

N = ⟨⟨αδ⟩⟩ · ⟨λ⟩.

The first formula for Q_{σ} then follows from (5) by Frobenius reciprocity. Since

⟨⟨αδ⟩⟩ = ⟨⟨α⟩⟩ · ⟨δ⟩ + ⟨⟨δ⟩⟩ in W F,

the following relation in W F follows:

Q_{σ} = ⟨1, 2, 2δ⟩ + ⟨2δ⟩ · ⟨⟨α⟩⟩ · t_{L/F} (⟨λ⟩) + ⟨2⟩ · ⟨⟨δ⟩⟩ · t_{L/F} (⟨λ⟩).

Lemma 10 shows that the last term on the right-hand side is equal to ⟨2⟩ · ⟨⟨δ⟩⟩. Since

⟨2δ⟩ + ⟨2⟩ · ⟨⟨δ⟩⟩ = ⟨2⟩ and ⟨1, 2, 2⟩ = ⟨1, 1, 1⟩,

we get

Q_{σ} = ⟨1, 1, 1⟩ + ⟨2δ⟩ · ⟨⟨α⟩⟩ · t_{L/F} (⟨λ⟩) in W F.

Since both sides have the same dimension, these two quadratic forms are isometric, proving the second formula for Q_{σ}.

Corollary 12. The form q_{σ} defined in Proposition 4 may be chosen as

q_{σ} = ⟨δ⟩ · t_{L/F} (⟨λ⟩)

for some λ ∈ L^{×} such that n_{L/F} (λ) ∈ F^{×}^{2}.

Proof. According to [16, p. 51], we have

det(t_{L/F} (⟨λ⟩)) = δ n_{L/F} (λ).

So far, the involution σ has been fixed, as well as the étale subalgebra L ⊂ (B, σ)_{+}. In the last proposition, we compare the quadratic forms Q_{σ} and Q_{σ′} associated to two involutions of the second kind which leave L elementwise invariant. We denote by

q: L → L

the quadratic map such that ℓq(ℓ) = n_{L/F} (ℓ) for all ℓ ∈ L.

Proposition 13. Let σ, σ′ be two involutions on B such that σ|_{L} = σ′|_{L} = I_{L}:

σ′ = Int(z) ∘ σ

for some z ∈ L^{×}. If λ ∈ L^{×} is such that

Q_{σ} = ⟨1, 2, 2δ⟩ ⊥ ⟨2⟩ · ⟨⟨αδ⟩⟩ · t_{L/F} (⟨λ⟩),

then

Q_{σ′} = ⟨1, 2, 2δ⟩ ⊥ ⟨2⟩ · ⟨⟨αδ⟩⟩ · t_{L/F} (⟨q(z)λ⟩).

Proof. The map b ↦ zb is an isomorphism (B, σ)_{+} → (B, σ′)_{+} which maps L to L; therefore, it also maps the orthogonal V of L in (B, σ)_{+} to the orthogonal V′ of L in (B, σ′)_{+}. Let N′: V′ → L denote the quadratic form

N′(v′) = \frac{1}{2} Trd(v^{′2}) − p′_{L}(v^{′2}) for v′ ∈ V′,

where p′_{L}: (B, σ′)_{+} → L denotes the orthogonal projection. An explicit computation, after extending scalars to an algebraic closure of F, shows that

N′(zv) = q(z)N(v) for all v ∈ V.

Therefore, multiplication by z defines a similarity N →^{∼} N′ with similarity factor q(z).

Proposition 13 leads to the following converse of Proposition 11:

Corollary 14. Let L be an arbitrary cubic étale F -subalgebra in B. For every λ ∈ L^{×} such that n_{L/F} (λ) ∈ F^{×}^{2}, there is an involution σ on B leaving L elementwise invariant such that

Q_{σ} = ⟨1, 2, 2δ⟩ ⊥ ⟨2⟩ · ⟨⟨αδ⟩⟩ · t_{L/F} (⟨λ⟩)
= ⟨1, 1, 1⟩ ⊥ ⟨2δ⟩ · ⟨⟨α⟩⟩ · t_{L/F} (⟨λ⟩).

Proof. By a theorem of Albert [1, p. 157], there is an involution τ leaving L elementwise invariant. Let

Q_{τ} = ⟨1, 2, 2δ⟩ ⊥ ⟨2⟩ · ⟨⟨αδ⟩⟩ · t_{L/F} (⟨μ⟩)

for some μ ∈ L^{×} such that n_{L/F} (μ) ∈ F^{×2}. If λ ∈ L^{×} is such that n_{L/F} (λ) ∈ F^{×}^{2}, let ξ ∈ F^{×} be such that

n_{L/F} (λμ^{−1}) = ξ^{2}.

Since n_{L/F} (λμ^{−1}) = λμ^{−1}q(λμ^{−1}), the preceding equality yields:

λμ^{−1} = ξ^{2}q(λμ^{−1})^{−1} = q(ξλ^{−1}μ).

Therefore, the preceding proposition shows that σ = Int(ξλ^{−1}μ) ∘ τ satisfies the required properties.

4. A Classification of Involutions in Degree 3. In this section we continue to assume that B has degree 3 and that F has characteristic different from 2, 3. By Proposition 4, the trace form of any involution σ of the second kind has the form

Q_{σ} = ⟨1, 1, 1⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · ⟨−b, −c, bc⟩

where K = F (\sqrt{α}) and b, c ∈ F. Our goal is to show that the 3-fold Pfister form ⟨⟨α, b, c⟩⟩ determines the involution σ up to isomorphism and to characterize the involutions for which this 3-fold Pfister form is hyperbolic.

For any Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩, we let ⟨⟨a_{1}, …, a_{n}⟩⟩^{♯} = ⟨1⟩^{⊥}, so that

⟨⟨b, c⟩⟩^{♯} = ⟨−b, −c, bc⟩.

Theorem 15. Let σ, σ′ be involutions of the second kind on a central simple K–algebra B of degree 3. Let

Q_{σ} = ⟨1, 1, 1⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · ⟨⟨b, c⟩⟩^{♯}

and

Q_{σ′} = ⟨1, 1, 1⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · ⟨⟨b′, c′⟩⟩^{♯}.

The following conditions are equivalent:

(a) The involutions σ and σ′ are isomorphic.

(b) The quadratic forms Q_{σ} and Q_{σ′} are isometric.

(c) The quadratic forms ⟨⟨α⟩⟩ · ⟨⟨b, c⟩⟩^{♯} and ⟨⟨α⟩⟩ · ⟨⟨b′, c′⟩⟩^{♯} are isometric.

(d) Either K = F ×F or the K-hermitian forms ⟨−b, −c, bc⟩_{K} and ⟨−b′, −c′, b′c′⟩_{K} are isometric.

(e) The quadratic forms ⟨⟨α, b, c⟩⟩ and ⟨⟨α, b′, c′⟩⟩ are isometric.

Proof. (a) ⇒ (b) is already in Lemma 1, and (b) ⟺ (c) ⟺ (e) follows by Witt cancellation. Moreover, (c) ⇒ (d) is a theorem of Jacobson [7] if K is a field, and is clear if K = F × F. We finally check that (d) ⇒ (a). If K = F × F, all involutions on B ≃ A × A^{op} are isomorphic to the switch involution, so (a) holds trivially. Thus we are reduced to the case K a field. Assume next that B = M_{3}(K) is split. Up to automorphisms of (B, σ), resp. (B, σ′), we have σ = Int(a)∘τ, resp. σ′ = Int(a′)∘τ with a = diag(α_{1}, α_{2}, α_{3}) and a′ = diag(α′_{1}, α′_{2}, α′_{3}). We may assume that α_{1}α_{2}α_{3} = 1 = α′_{1}α′_{2}α′_{3}. By Proposition 2, we have

Q_{σ} ≃ ⟨1, 1, 1⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · ⟨α_{2}α_{3}^{−1}, α_{3}α_{1}^{−1}, α_{1}α_{2}^{−1}⟩

and

Q_{σ′} ≃ ⟨1, 1, 1⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · ⟨α′_{2}α_{3}^{′−1}, α′_{3}α′_{1}^{−1}, α′_{1}α′_{2}^{−1}⟩.

Thus

⟨α_{2}α_{3}^{−1}, α_{3}α_{1}^{−1}, α_{1}α_{2}^{−1}⟩_{K} ≃ ⟨α′_{2}α′_{3}^{−1}, α′_{3}α′_{1}^{−1}, α′_{1}α′_{2}^{−1}⟩_{K}.

Since

⟨α_{1}α_{2}α_{3}⟩ · ⟨α_{2}α_{3}^{−1}, α_{3}α_{1}^{−1}, α_{1}α_{2}^{−1}⟩_{K} ≃ ⟨α_{1}, α_{2}, α_{3}⟩_{K},

we get isometries h_{a} ≃ h_{a′} or ⟨a⟩_{M_{3}(K)} ≃ ⟨a′⟩_{M_{3}(K)} and (M_{3}(K), σ) is isomorphic to (M_{3}(K), σ′) by Lemma 1.

If B is not split and if σ′ = Int(u)∘σ, we have to check by Lemma 1 that the hermitian spaces ⟨u⟩_{B} and ⟨1⟩_{B} are similar. Replacing u by u · Nrd(u), we may assume that Nrd(u) ∈ F^{×}^{2}. Let L = F (x) with x ∈ (B, σ)_{+}, x ∉ F, so that L is a field extension of F of degree 3. The algebra B ⊗_{F} L is split over K ⊗_{F} L^{×} and there is v′ ∈ GL_{3}(K ⊗_{F} L) such that

u ⊗ 1 = λv′(σ ⊗ 1)(v′)

Thus, denoting by : K → K the non-trivial automorphism of K/F,

Nrd(u ⊗ 1) = λ^{3} Nrd(v′)\overline{Nrd(v′)} = μ^{2}

with μ ∈ F, since Nrd(u) ∈ F^{×}^{2}, and we can write

λ = (μλ^{−1} Nrd(v′)^{−1})(\overline{μλ^{−1} Nrd(v′)^{−1}}) = ν\overline{ν}.

It follows that u ⊗ 1 = v(σ ⊗ 1)(v) with v = νv′, so ⟨u⟩_{B⊗L} ≃ ⟨1⟩_{B⊗L}. By the Bayer–Lenstra [4] generalization of Springer’s theorem to hermitian spaces, we have ⟨u⟩_{B} ≃ ⟨1⟩_{B} and, by Lemma 1, (B, σ) ≃ (B, σ′).

In view of the equivalence (a) ⟺ (e) in Theorem 15, the Pfister form ⟨⟨α, b, c⟩⟩ classifies involutions σ on B. We denote it by π(B, σ). Note that π(B, σ) is isometric to the norm of the octonion algebra OctA associated in [14] to any simple Jordan algebra A of degree 3 and dimension 9, if A = (B, σ)_{+}. Let

(B, σ)_{+}^{∘} = {x ∈ (B, σ)_{+}: Trd(x) = 0} = 1^{⊥} ⊂ (B, σ)_{+}.

Since Q_{σ}(1) = 3, the restriction Q_{σ}^{∘} of Q_{σ} to (B, σ)_{+}^{∘} is given by

Q_{σ}^{∘} ≃ ⟨2, 6⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · ⟨⟨b, c⟩⟩^{♯} ≃ ⟨2⟩ · (⟨1, 3⟩ ⊥ ⟨⟨α⟩⟩ · ⟨⟨b, c⟩⟩^{♯}).

In the Witt ring W F, we have

⟨⟨α⟩⟩ · ⟨⟨b, c⟩⟩^{♯} = ⟨⟨α, b, c⟩⟩ − ⟨⟨α⟩⟩;

therefore,

Q_{σ}^{∘} = ⟨2⟩ · (⟨3, α⟩ + ⟨⟨α, b, c⟩⟩) in W F.

Comparing dimensions on both sides, we see that the Witt indices of both sides are related by:

w(Q_{σ}^{∘}) = w(⟨3, α⟩ ⊥ ⟨⟨α, b, c⟩⟩) − 1.  (9)

Isotropic elements of Q_{σ}^{∘} are elements u ∈ (B, σ)_{+} such that Trd(u) = Trd(u^{2}) = 0. Since the reduced characteristic polynomial of any a ∈ B has the form

X^{3} − Trd(a)X^{2} + \frac{1}{2} [Trd(a)^{2} − Trd(a^{2})] X − Nrd(a) · 1, it follows that u is isotropic if and only if u^{3} = Nrd(u) ∈ F. Theorem 16. The following conditions are equivalent:

(a) π(B, σ) is hyperbolic;

(b) Either K = F × F or ⟨−b, −c, bc⟩_{K} ≃ ⟨1, −1, −1⟩_{K};

(c) w(Q_{σ}^{∘}) ≥ 2; (c’) (B, σ)_{+} contains a subspace U of dimension 2 whose elements

satisfy: u^{3} = Nrd(u);

(d) w(Q_{σ}^{∘}) ≥ 3; (d’) (B, σ)_{+} contains a subspace U of dimension 3 whose elements

satisfy: u^{3} = Nrd(u);

(e) (B, σ)_{+} contains an étale cubic F -algebra L of discriminant α.

(f) B is a crossed product:

B = M ⊕ M x ⊕ M x^{2}, where M (⊃ K) is a Galois extension of F with Galois group S_{3}, the involution σ preserves M and leaves x invariant.

Proof. (a) ⇒ (b) is a straightforward consequence of Theorem 15, and (c) ⟺ (c’), (d) ⟺ (d’) follow from the preceding observations on isotropic elements in Q_{σ}^{∘}. Moreover, (a) ⇒ (d) follows from (9),

(e) ⟺ (f) from Proposition 8, and (d) ⇒ (c) is clear. We now show (c) ⇒ (a): if the Witt index of Q_{σ}^{∘} is at least 2, then (9) shows that ⟨3, α⟩ ⊥ ⟨⟨α, b, c⟩⟩ contains isotropic subspaces of dimension 3. Therefore, ⟨⟨α, b, c⟩⟩ is isotropic, hence hyperbolic, proving (a). Assuming (e), the formula for Q_{σ} in Proposition 11 shows that w(Q_{σ}^{∘}) ≥ 3, hence (e) ⇒ (d). To complete the proof, we show (c’) ⇒ (e), using results from [6]. We first consider the easy special case where B is not a division algebra. If α = 1 in F^{×}/F^{×}^{2}, then we may assume

B = A × A^{op}

for some central simple F -algebra A of degree 3, and σ is the switch involution. A theorem of Wedderburn [1] shows that A contains a cyclic extension L of F. We may then choose

M = {(ℓ, ℓ^{op}): ℓ ∈ L}.

If B is split, then since (c’) ⇒ (b), it follows that σ is the adjoint involution with respect to an isotropic hermitian form. We may thus

assume B = M_{3}(K) and σ = Int(u) ∘ τ, where u = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ). We

may then choose

M = \{( \begin{matrix}f & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & \overline{k} \end{matrix} ): f ∈ F, k ∈ K\} .

For the rest of the proof of (c’) ⇒ (e), we now assume that B is a division algebra. Let U ⊂ (B, σ)_{+} be a subspace of dimension 2 such that u^{3} = Nrd(u) for all u ∈ U. According to Lemma 2 of [6], one can find a basis (w_{1}, w_{2}) of U such that

Trd(w_{1}^{−1}w_{2}) = 0.

Following [6], we let

θ_{1} = w_{1}^{−1}w_{2},  θ_{2} = w_{1}^{−1}θ_{1}w_{1},  θ_{3} = w_{1}^{−1}θ_{2}w_{1}.

Note θ_{2} = w_{1}^{−3}(w_{1}w_{2}w_{1}), so σ(θ_{2}) = θ_{2}. We let also E = K(θ_{2}^{−1}θ_{3}) if θ_{2}^{−1}θ_{3} ∉ K and E = K(θ_{2}) if θ_{2}^{−1}θ_{3} ∈ K. Theorem 3 of [6] shows that E/K is cyclic and E/F is Galois with Galois group S_{3}. Moreover, one of the order 2 automorphisms of E/F is the restriction of the involution

σ′ = Int(θ_{2}^{−1}) ∘ σ.

Let L′ ⊂ E denote the subfield of σ′-invariant elements. Since L′K = E is cyclic over K and Galois over F with group S_{3}, Proposition 8 shows that the discriminant of L′/F is α.

Observe now that

θ_{2} = w_{1}^{−3}w_{2}^{−3}(w_{1}w_{2}^{2})(w_{2}^{2}w_{1}),

hence θ_{2}^{−1} = λvσ(v) for λ = w_{1}^{3}w_{2}^{3} and v = w_{1}^{−1}w_{2}^{−2}, so that

Int(v): (B, σ) →^{∼} (B, σ′)

is an isomorphism of algebras with involution. Pulling L′ ⊂ (B, σ′)_{+} back gives the wanted extension L ⊂ (B, σ)_{+}.

Remark. Theorem 16 gives a positive answer to a question about Tits constructions asked in [14, (2.12)].

An involution σ satisfying the equivalent conditions of Theorem 16 is called distinguished. By Theorem 15, two distinguished involutions on B are isomorphic. The existence of distinguished involutions is clear if B ≃ M_{3}(K) is split: the adjoint involution with respect to any isotropic hermitian form on K^{3} is distinguished. If K = F × F, the switch involution on B ≃ A × A^{op} is distinguished: in fact we have α = 1, so that ⟨⟨α⟩⟩ = ⟨1, −1⟩ and Q_{σ}^{∘} has Witt index at least 3. The existence in general of distinguished involutions for algebras of degree 3 is shown next.

Proposition 17. For every cubic étale F -algebra L ⊂ B, there is a distinguished involution σ such that L ⊂ (B, σ)_{+}.

Proof. Let λ_{0} ∈ L^{×} be such that t_{L/F} (λ_{0}) = 0, and let

λ = λ_{0}/n_{L/F} (λ_{0});

then n_{L/F} (λ) = n_{L/F} (λ_{0})^{−2} ∈ F^{×}^{2} and t_{L/F} (λ) = 0. Corollary 14 shows that there is an involution σ on B such that L ⊂ (B, σ)_{+} and

Q_{σ} = ⟨1, 1, 1⟩ ⊥ ⟨2δ⟩ · ⟨⟨α⟩⟩ · t_{L/F} (⟨λ⟩),

hence

Q_{σ}^{∘} = ⟨2⟩ · (⟨1, 3⟩ ⊥ ⟨δ⟩ · ⟨⟨α⟩⟩ · t_{L/F} (⟨λ⟩)).

Since t_{L/F} (λ) = 0, the form t_{L/F} (⟨λ⟩) is isotropic, hence the Witt index of ⟨⟨α⟩⟩ · t_{L/F} (⟨λ⟩) is at least 2. Therefore, condition (c) of Theorem 16 holds.

Alternative proof. Pick an involution σ_{0} such that L ⊂ (B, σ_{0})_{+} and denote by V the orthogonal of L with respect to Q_{σ_{0}}, as in the preceding section. By Zariski density, we may find an invertible element v ∈ V such that n_{L/F} (N(v)) ≠ 0, where N: V → L is the quadratic form defined in section 3. Lemma 6 shows:

Trd(v^{−1}ℓ) = − Nrd(v)^{−1}t_{L/F} (N(v)ℓ)

for all ℓ ∈ L. Since N(v) is invertible in L, we may, again by Zariski density, find ℓ ∈ L^{×} such that

Trd(v^{−1}ℓ) = 0.

Following [6, Proposition 1], the vector space

U = v^{−1}L ∩ ker Trd

is at least 2-dimensional over F and satisfies: u^{3} = Nrd(u) for all u ∈ U. We then have v^{−1}ℓ ∈ U for ℓ ∈ L as above. Let

σ = Int(ℓ^{−1}) ∘ σ_{0}.

Since L ⊂ (B, σ_{0})_{+} and ℓ ∈ L^{×}, it follows that L ⊂ (B, σ)_{+}. We claim that σ is distinguished. Let y = ℓ^{−1}v ∈ B^{×}, then

yU = ℓ^{−1}(vU) ⊂ L ⊂ (B, σ)_{+}

and

σ(y) = ℓ^{−1}σ_{0}(y)ℓ = ℓ^{−1}vℓ^{−1}ℓ = y,

hence (B, σ)_{+} contains y^{2}Uy = y(yU)y. Since v^{−1}ℓ ∈ U, we have y^{−1} ∈ U, hence y^{3} = Nrd(y) ∈ F^{×}. For u ∈ U, we have u^{3} = Nrd(u) ∈ F, hence

(y^{2}uy)^{3} = y^{2}uy · y^{2}uy · y^{2}uy = y^{2}u^{3}y Nrd(y)^{2}
= Nrd(u) Nrd(y)^{3} = Nrd(y^{2}uy).

Therefore, y^{2}Uy ⊂ (B, σ)_{+} is a subspace of dimension ≥ 2 whose elements satisfy x^{3} = Nrd(x), hence σ is distinguished.

A third proof of the existence of distinguished involutions is given in Proposition 31.

Corollary 18. The space (B, σ)_{+} contains an isomorphic copy of every cubic étale F -subalgebra L of B if and only if σ is distinguished.

Proof. Since all the distinguished involutions on a central simple algebra B are isomorphic, the if direction follows from Proposition 17. Conversely, by Theorem 16, (e), the only involutions which leave elementwise invariant étale cubic F -subalgebras of discriminant α are the distinguished involutions.

The fact that the only involutions which leave elementwise invariant étale cubic F -subalgebras of discriminant α are the distinguished involutions also follows from the following general result:

Proposition 19. Let σ be an involution on B and let L be a cubic étale F -algebra such that L ⊂ (B, σ)_{+}. Let δ ∈ F^{×} represent the discriminant of L/F. The Pfister form π(B, σ) has a factorization:

π(B, σ) = ⟨⟨α⟩⟩ · φ

where φ is a 2-fold Pfister form whose pure subform satisfies:

φ^{♯} = t_{L/F} (⟨ℓ⟩)

for some ℓ ∈ L^{×} such that

n_{L/F} (ℓ) ∈ δ · F^{×}^{2}.

In particular,

φ · ⟨⟨δ⟩⟩ = 0.

Proof. Proposition 11 shows that

Q_{σ} = ⟨1, 1, 1⟩ ⊥ ⟨2⟩ · ⟨⟨α⟩⟩ · t_{L/F} (⟨δλ⟩)

where λ ∈ L^{×} is such that n_{L/F} (λ) ∈ F^{×}^{2}. Letting ℓ = δλ, we get n_{L/F} (ℓ) ∈ δ · F^{×}^{2} and t_{L/F} (⟨ℓ⟩) is a 3-dimensional form of determinant 1. Comparing with the form of Q_{σ} in Theorem 15, we obtain:

π(B, σ) = ⟨⟨α⟩⟩ · (⟨1⟩ ⊥ t_{L/F} (⟨ℓ⟩)).

The relation (⟨1⟩ ⊥ t_{L/F} (⟨δλ⟩)) · ⟨⟨δ⟩⟩ = 0 follows from Lemma 10, using the fact that ⟨−δ⟩ · ⟨⟨δ⟩⟩ = ⟨⟨δ⟩⟩.

The condition φ · ⟨⟨δ⟩⟩ = 0 in the proposition above yields some restriction on the discriminants of cubic F -algebras L which lie in (B, σ)_{+}. It does not yield any information on cyclic extensions however, since in this case ⟨⟨δ⟩⟩ = 0. Nevertheless, there are examples where (B, σ)_{+} does not contain any cyclic cubic extension of F. Consider for instance the “symbol algebra” A_{ω}(s, t) generated over the iterated powerseries field _{ℂ}((s))((t)) by two indeterminates i, j subject to the relations:

i^{3} = s,  j^{3} = t,  ji = ωij,

where ω is a primitive cube root of unity. This algebra carries an involution σ extending conjugation on _{ℂ} such that

σ(i) = i, σ(j) = j.

The subfield consisting of the σ-invariant elements in the center ℂ((s))((t)) is _{ℝ}((s))((t)), which does not have any non-trivial cyclic extension of odd degree. Therefore, the space of symmetric elements (A_{ω}(s, t), σ)_{+} does not contain any cyclic extension of F of degree 3. We are indebted to H.P. Petersson (see [13]) for suggesting this example.

Let L be an arbitrary cubic étale F -subalgebra in B. Let Σ(B, L) be the pointed set of isomorphism classes of involutions σ of B such that σ|_{L} = I_{L}, the point being given by the class of distinguished involutions.

Corollary 20. The map μ ∈ L^{×} ↦ σ_{μ}, where σ_{μ} is the involution determined up to isomorphism by the trace form

Q_{σ_{μ}} = ⟨1, 1, 1⟩ ⊥ ⟨2δ⟩ · ⟨⟨α⟩⟩ · t_{L/F} (⟨q(μ)⟩_{L}),

induces a surjective map of pointed sets

L^{×}/n_{LK/L}(LK^{×}) · F^{×} → Σ(B, L).

If L/F is cyclic, the map is well-defined on the set of orbits of the group Gal(L/F) in

L^{×}/n_{LK/L}(LK^{×}) · F^{×}.

Proof. The proof of Corollary 14 shows that the elements λ ∈ L such that n_{L/F} (λ) ∈ F^{×}^{2} are exactly the elements of the form q(μ) for μ ∈ L^{×}. Thus, by Corollary 14, Q_{σ_{μ}} is the trace form of an involution σ_{μ}. Elements μ, μ′ ∈ L^{×} such that μ′ = ξn_{LK/L}(η)μ for η ∈ LK^{×} and ξ ∈ F^{×} give isomorphic involutions in view of Theorem 15, (d), since q(μ′) = ξ^{2} q(η) q(μ) \overline{q(η)} implies that the hermitian forms ⟨q(μ)⟩_{LK} and ⟨q(μ′)⟩_{LK} are isomorphic.

5. Cohomology and Symbols. Let F be a field of characteristic different from 2; let F_{s} denote a separable closure of F and Γ = Gal(F_{s}/F). For any integer n relatively prime to the characteristic of F, let

μ_{n} = {x ∈ F_{s}: x^{n} = 1}

denote the Γ-module of n-th roots of 1. The Kummer exact sequence

1 → μ_{n} → F_{s}^{×} →^{n} F_{s}^{×} → 1

and Hilbert’s Theorem 90 yield canonical isomorphisms:

H^{1}(Γ, μ_{n}) = F^{×}/F^{×n} and H^{2}(Γ, μ_{n}) = _{n}Br(F),

where _{n}Br(F) denotes the subgroup killed by n in the Brauer group Br(F). Since the action of Γ on μ_{2} is trivial, we also have

H^{1}(Γ, μ_{2}) = Hom(Γ, {±1}).

For any α ∈ F^{×}, let ϕ_{α}: Γ → {±1} be the (continuous) homomorphism corresponding to α · F^{×}^{2} under the identification: F^{×}/F^{×}^{2} = Hom(Γ, {±1}); it is explicitly defined by:

ϕ_{α}(γ) = \{ \begin{matrix}+1 & if γ leaves invariant \\ −1 & if γ permutes \end{matrix} \} the square roots of α in F_{s}.

We then define a Γ-module _{ℤ}(α) by twisting the trivial action of Γ on _{ℤ}: for γ ∈ Γ and z ∈ _{ℤ}, we let γ · z = ϕ_{α}(γ)z. Multiplication in _{ℤ} yields a canonical isomorphism:

ℤ(α) ⊗_{ℤ} _{ℤ}(β) = _{ℤ}(αβ) for α, β ∈ F^{×},

since ϕ_{α}(γ)ϕ_{β}(γ) = ϕ_{αβ}(γ) for all γ ∈ Γ.

For any Γ-module M, we define a twisted module M(α) by:

M(α) = M ⊗_{ℤ} _{ℤ}(α).

The Galois cohomology groups H^{ℓ}(Γ, M) are also denoted H^{ℓ}(F, M).

Proposition 21. Let K = F (\sqrt{α}) be a quadratic field extension of F. For all ℓ ≥ 1 and all Γ-modules M on which multiplication by 2 is invertible, the following sequence is split exact:

0 → H^{ℓ}(F, M(α)) ⟶^{res} H^{ℓ}(K, M) ⟶^{cor} H^{ℓ}(F, M) → 0

where res (resp. cor) is the restriction (resp. the corestriction) map.

The restriction map identifies H^{ℓ}(F, M(α)) with the subgroup of H^{ℓ}(K, M) consisting of the classes of cocycles which are cohomologous to the negative of their conjugate under the action of Gal(K/F):

H^{ℓ}(F, M(α)) = \{φ ∈ H^{ℓ}(K, M): φ + \overline{φ} = 0\}.

Proof. From the definition of M(α), it is clear that M = M(α) as modules over Gal(F_{s}/K). Let G = Gal(K/F) = {1, ι} and let IG = _{ℤ} ·(1 − ι) denote the augmentation ideal in the group ring _{ℤ} G. Denote by Σ the augmentation map:

Σ: _{ℤ} G → _{ℤ}

and define Γ-module homomorphisms s: _{ℤ} → _{ℤ} G and t: _{ℤ} G → IG by:

s(z) = z(1 + ι), t(x) = (1 − ι)x for z ∈ _{ℤ}, x ∈ _{ℤ} G.

These maps fit into a commutative diagram:

0 ← IG ⟵^{t} _{ℤ} G ⟵^{s} _{ℤ} ← 0
↑^{2·}  ∥  ↓ ^{2·}
0 → IG ⟶ _{ℤ} G ⟶^{Σ} _{ℤ} → 0

Observe that IG ≃ _{ℤ}(α); therefore, tensoring the diagram above with M, we get the following commutative diagram:

0 ← M(α) ⟵^{t} M ⊗_{ℤ} _{ℤ} G ⟵^{s} M
↑^{2·}  ∥  ↓ ^{2·}
M(α) ⟶^{i} M ⊗_{ℤ} _{ℤ} G ⟶^{Σ} M → 0

and since multiplication by 2 is invertible on M, it follows that the sequence

0 → M(α) ⟶^{i} M ⊗_{ℤ} _{ℤ} G ⟶^{Σ} M → 0

is split exact. It yields a split exact sequence in cohomology:

0 → H^{ℓ}(F, M(α)) ⟶^{i_{∗}} H^{ℓ}(F, M ⊗_{ℤ} G) ⟶^{Σ_{∗}} H^{ℓ}(F, M) → 0.

By the lemma of Eckmann–Faddeev–Shapiro, there is an isomorphism

ψ: H^{ℓ}(F, M ⊗_{ℤ} G) →^{∼} H^{ℓ}(K, M)

such that Σ_{∗} = cor ∘ ψ and ψ ∘ i_{∗} = res. This proves the first part.

Let res′: H^{ℓ}(F, M) → H^{ℓ}(K, M) denote the restriction map. For φ ∈ H^{ℓ}(K, M), we have:

res′ ∘ cor(φ) = φ + \overline{φ},  (10)

hence φ+\overline{φ} = 0 if cor(φ) = 0. Conversely, applying cor to both sides of (10), we get:

2 cor(φ) = cor(φ + \overline{φ}),

since cor ∘ res′ = 2. Since multiplication by 2 is invertible on M, it follows that cor(φ) = 0 if φ + \overline{φ} = 0. Therefore, the image of H^{ℓ}(F, M(α)) in H^{ℓ}(K, M) can be indifferently described as the kernel of the corestriction map or as the kernel of the map φ ↦ φ + \overline{φ}.

Remark. If multiplication by 2 is not invertible on M, the restriction and corestriction maps fit into a long exact sequence described by Arason and Elman in [3, Appendix].

We shall apply this proposition to the special cases where M = ℤ /n_{ℤ} (n odd) with trivial action and M = μ_{n} = {x ∈ F_{s}: x^{n} = 1} (assuming n odd and relatively prime to the characteristic of F). We first review how, for an arbitrary finite group G, the first cohomology set H^{1}(F, G), where the Galois action on G is trivial, is related to Galois extensions of F with Galois group G. We recall that, in this case,

H^{1}(F, G) = Hom(Γ, G)/ ∼,

where Hom(Γ, G) is the set of continous homomorphisms Γ → G and χ′ ∼ χ if χ′(γ) = gχ(γ)g^{−1} for some g ∈ G. In particular we have H^{1}(F, G) = Hom(Γ, G) if G is abelian.

Let L be a finite-dimensional commutative algebra over F and let

X(L) = Alg_{F} (L, F_{s})

denote the set of F -algebra homomorphisms L → F_{s}. Let G be an arbitrary finite group acting on L by F -algebra automorphisms ℓ ↦ g ∗ ℓ. We call L a Galois G-algebra if L is étale of dimension n = |G| over F and the action of G on X(L) is simply transitive.

Now, assume L is a Galois G-algebra and fix some ξ ∈ X(L). For every γ ∈ Γ, there is a unique χ_{ξ}(γ) ∈ G such that

γ(ξ(ℓ)) = ξ(χ_{ξ}(γ) ∗ ℓ) for all ℓ ∈ L.

The map χ_{ξ}: Γ → G is a continuous homomorphism:

χ_{ξ} ∈ Hom(Γ, G).

If L, L′ are Galois G-algebras over F and ξ ∈ X(L), ξ′ ∈ X(L′), an isomorphism ψ: (L, ξ) →^{∼} (L′, ξ′) is an isomorphism of algebras over F which commutes with the action of G and such that ξ′ ∘ ψ = ξ. An isomorphism of Galois G-algebras ψ: L →^{∼} L′ is an isomorphism of algebras over F which commutes with the action of G.

Proposition 22. The map (L, ξ) ↦ χ_{ξ} defines a 1–1 correspondence between the set of isomorphism classes of couples (L, ξ), where L is a Galois G-algebra and ξ ∈ X(L), and the set Hom(Γ, G). Moreover L ↦ χ_{ξ} induces a 1–1 correspondence between the set of isomorphism classes of Galois G-algebras and the set H^{1}(F, G).

Proof. This proposition is presumably well-known (see for example [17]), but we include a sketch of proof for the reader’s convenience. For χ ∈ Hom(Γ, G), we define an action of Γ on the algebra 𝔉(G, F_{s}) of maps G → F_{s} by:

(γ ⋄ f)(x) = γ(f(χ(γ)^{−1}x)) for γ ∈ Γ, x ∈ G.

Let 𝔉_{χ} denote the algebra of invariant elements under this action:

𝔉_{χ} = {f: G → F_{s}: γ(f(x)) = f(χ(γ)x) for γ ∈ Γ, x ∈ G}.

This algebra carries a natural G-action defined by:

(g ∗ f)(x) = f(xg) for x, g ∈ G, f ∈ 𝔉_{χ}.

Moreover, for g ∈ G, there is an F -algebra homomorphism

ε_{g}: 𝔉_{χ} → F_{s}

defined by:

ε_{g}(f) = f(g) for f ∈ 𝔉_{χ},

and X(𝔉_{χ}) = {ε_{g}: g ∈ G}. It is then straightforward to check that 𝔉_{χ} is a Galois G-algebra, and that χ_{ε_{1}} = χ.

For any Galois G-algebra L and any ξ ∈ X(L), there exists a unique isomorphism (L, ξ) →^{∼} (𝔉_{χ_{ξ}}, ε_{1}): this isomorphism maps ℓ ∈ L to f_{ℓ} ∈ 𝔉_{χ} defined by:

f_{ℓ}(g) = ξ(g ∗ ℓ) for g ∈ G.

Let now ξ, η ∈ X(L). By definition of Galois algebra, there is a unique element g ∈ G such that η(ℓ) = ξ(g ∗ ℓ) for all ℓ ∈ L. Then, for γ ∈ Γ and ℓ ∈ L,

γ(η(ℓ)) = γ(ξ(g ∗ ℓ)) = ξ(χ_{ξ}(γ)g ∗ ℓ) = ξ(g ∗ (g^{−1}χ_{ξ}(γ)g ∗ ℓ)),

hence

χ_{η}(γ) = g^{−1}χ_{ξ}(γ)g.

This implies the last claim of Proposition 22.

We now consider the particular case where G is dihedral: let D_{n} be the group generated by two elements r, s subject to the relations:

r^{n} = 1, s^{2} = 1 and rsr = s,

and let Z_{n} denote the cyclic (normal) subgroup of D_{n} generated by r. The group D_{n} is the semidirect product D_{n} = Z_{n} ⋊ μ_{2} and we have a split exact sequence

1 → Z_{n} → D_{n} ⟶^{ρ} μ_{2} → 1.  (11)

Proposition 23. Let L be a Galois D_{n}-algebra over F and let

L_{0} = {ℓ ∈ L: r ∗ ℓ = ℓ}, L_{1} = {ℓ ∈ L: s ∗ ℓ = ℓ}.

Then L_{0} ≃ F (\sqrt{α}) for some α ∈ F^{×}. For all ξ ∈ X(L), the homomorphism ρ ∘ χ_{ξ} ∈ Hom(Γ, μ_{2}) corresponds to α · F^{×}^{2} under the identification Hom(Γ, μ_{2}) = H^{1}(F, μ_{2}) = F^{×}/F^{×}^{2}. Moreover, if n is odd, α · F^{×}^{2} is the discriminant of L/F, so that the map H^{1}(F, D_{n}) → H^{1}(F, μ_{2}), induced in cohomology by ρ, associates to L the discriminant of L. The discriminant of L_{1}/F is 1 if n ≡ 1 mod 4 and α · F^{×2} if n ≡ 3 mod 4.

Proof. The first statement is clear, since Galois theory shows L_{0} is a quadratic étale algebra over F. Since the action of Z_{n} on L_{0} is trivial, the action of D_{n} factors through μ_{2}: we may set

ρ(g) ∗ ℓ = g ∗ ℓ for g ∈ D_{n} and ℓ ∈ L_{0}.

For all ξ ∈ X(L), we then have

γ(ξ(ℓ)) = ξ(ρ ∘ χ_{ξ}(γ) ∗ ℓ) for γ ∈ Γ and ℓ ∈ L_{0},

proving that ρ ∘ χ_{ξ} ∈ H^{1}(F, μ_{2}) is the homomorphism corresponding to the Galois μ_{2}-algebra L_{0}/F.

For the rest of the proof, assume n is odd. The discriminant of L_{1}/F is represented by the determinant

det(tr_{L_{1}/F} (e_{i}e_{j}))_{1≤i,j≤n},

where (e_{i})_{1≤i≤n} is a basis of L_{1} over F. Since, for ℓ ∈ L_{1},

tr_{L_{1}/F} (ℓ) = ∑_{i=1}^{n} ξ(r^{i} ∗ ℓ),

we have

(tr_{L_{1}/F} (e_{i}e_{j}))_{1≤i,j≤n} = m^{t} · m

where

m = (ξ(r^{i} ∗ e_{j}))_{1≤i,j≤n}.

Therefore, if δ_{1} = det m ∈ ξ(L), the discriminant of L_{1}/F is represented in F^{×}/F^{×}^{2} by δ_{1}^{2}. For γ ∈ Γ, we have

γ(δ_{1}) = ξ(det(χ_{ξ}(γ)r^{i} ∗ e_{j})_{1≤i,j≤n}) = ξ(det(χ_{ξ}(γ)r^{i}s ∗ e_{j})_{1≤i,j≤n}).

If χ_{ξ}(γ) ∈ Z_{n}, multiplication by χ_{ξ}(γ) is an even permutation of Z_{n}, hence

det(χ_{ξ}(γ)r^{i} ∗ e_{j})_{1≤i,j≤n} = det(r^{i} ∗ e_{j})_{1≤i,j≤n},

and therefore γ(δ_{1}) = δ_{1}. If χ_{ξ}(γ) ∉ Z_{n}, the map z ↦ χ_{ξ}(γ)zs is a permutation of Z_{n} of signature (−1)^{(n−1)/2}, hence

γ(δ_{1}) = (−1)^{(n−1)/2}δ_{1}.

Therefore, if n ≡ 1 mod 4 we have δ_{1} ∈ F, hence the discriminant of L_{1}/F is trivial. If n ≡ 3 mod 4, we have

γ(δ_{1}) = ρ ∘ χ_{ξ}(γ) δ_{1},

hence the discriminant of L_{1}/F corresponds to ρ ∘ χ_{ξ} ∈ H^{1}(F, μ_{2}), hence it is α · F^{×}^{2}.

The discriminant of L/F is calculated in a similar way: multiplication by g ∈ D_{n} defines on D_{n} a permutation of signature ρ(g), hence the discriminant of L/F corresponds to ρ ∘ χ_{ξ} ∈ H^{1}(F, μ_{2}).

Suppose now K = F (\sqrt{α}) is a quadratic field extension of F contained in F_{s}. We denote by Γ_{K} the Galois group Γ_{K} = Gal(F_{s}/K) of F_{s} over K.

Proposition 24. Suppose n is odd. The group H^{1}(F, Z_{n}(α)) classifies the Galois Z_{n}-algebras L over K which can be endowed with a Galois D_{n}-algebra structure over F extending the action of Z_{n}.

Proof. Let σ ∈ Γ ∖ Γ_{K}. Proposition 21 shows that

H^{1}(F, Z_{n}(α)) = \{χ ∈ H^{1}(K, Z_{n}): χ\overline{χ} = 1\}
= \{χ ∈ Hom(Γ_{K}, Z_{n}): χ\overline{χ} = 1\},

where \overline{χ} is defined by:

\overline{χ}(γ) = χ(σγσ^{−1}) for γ ∈ Γ_{K}.

Let L be a Galois Z_{n}-algebra over K which also has a Galois D_{n}- algebra structure over F extending the action of Z_{n}. Let ξ ∈ X(L). We may assume ξ is a K-algebra homomorphism; the restriction to Γ_{K} of the associated homomorphism χ_{ξ}: Γ → D_{n} is then the element χ_{ξ}|_{Γ_{K}} ∈ H^{1}(K, Z_{n}) associated to L, viewed as a Galois Z_{n}-algebra over K.

If χ_{ξ}(σ) ∈ Z_{n}, then ρ maps the image of χ_{ξ} to 1 ∈ μ_{2}, hence Proposition 23 shows that the subalgebra L_{0} of invariant elements under r splits as F × F. This is a contradiction since L_{0} = K. Therefore, χ_{ξ}(σ) ∉ Z_{n}, hence, for γ ∈ Γ_{K},

χ_{ξ}|_{Γ_{K}}(σγσ^{−1}) = χ_{ξ}(σ)χ_{ξ}(γ)χ_{ξ}(σ)^{−1} = χ_{ξ}|_{Γ_{K}}(γ)^{−1}.

This shows that χ_{ξ}|_{Γ_{K}} \overline{χ_{ξ}|_{Γ_{K}}} = 1, hence χ_{ξ}|_{Γ_{K}} ∈ H^{1}(F, Z_{n}(α)).

Conversely, to every χ ∈ H^{1}(F, Z_{n}(α)) we associate the Galois Z_{n}-algebra over K:

𝔉_{χ} = {f ∈ 𝔉(Z_{n}, F_{s}): γ(f(x)) = f(χ(γ)x) for all x ∈ Z_{n}}.

Since χ\overline{χ} = 1, we may extend the Z_{n}-action on 𝔉_{χ} to an action of D_{n} by letting:

(s ∗ f)(x) = σ(f(x^{−1})) for f ∈ 𝔉_{χ}, x ∈ Z_{n}.

Remark. Another proof of the proposition above can be obtained by defining a twisted action of Γ on D_{n} extending the action on Z_{n}. There is an exact sequence corresponding to (11):

1 → Z_{n}(α) → D_{n}(α) → μ_{2} → 1,

and the associated cohomology sequence:

1 → H^{1}(F, Z_{n}(α)) → H^{1}(F, D_{n}(α)) → H^{1}(F, μ_{2})

yields an alternative (equivalent) description of H^{1}(F, Z_{n}(α)).

As observed in the proof, all Galois Z_{n}-algebras L over K which can be endowed with a Galois D_{n}-algebra structure over F extending the Z_{n}-action have discriminant α · F^{×}^{2}. It is also clear from the proof that the D_{n}-algebra structure on L is not uniquely determined, since it depends on the choice of σ. However, the subalgebra L_{1} of invariant elements under the element s of D_{n} does not depend on the choice of the D_{n}-algebra structure up to F -isomorphism. Therefore, for χ ∈ H^{1}(F, Z_{n}(α)), we may denote by F_{χ} an F -algebra which is isomorphic to the subalgebra of s-invariant elements in the algebra L corresponding to χ under the correspondence of Proposition 24. According to Proposition 23, the discriminant of F_{χ} is 1 if n ≡ 1 mod 4 and α · F^{×}^{2} if n ≡ 3 mod 4.

Any χ ∈ H^{1}(F, Z_{n}) defines a Galois Z_{n}-algebra over F which we will also denote by F_{χ}. The discriminant of this extension is 1.

We now turn to the Galois module μ_{n} of n-th roots of unity, assuming that n is relatively prime to the characteristic of F. As above, we denote

μ_{n}(α) = μ_{n} ⊗_{ℤ} _{ℤ}(α).

Proposition 25. Let K = F (\sqrt{α}) be a quadratic field extension of F. For any odd integer n relatively prime to the characteristic of F,

H^{1}(F, μ_{n}(α)) = \{x · K^{×n} ∈ K^{×}/K^{×n}: x\overline{x} = 1\}

and H^{2}(F, μ_{n}(α)) classifies up to Brauer-equivalence central simple K-algebras of exponent n which admit involutions of the second kind leaving F elementwise invariant.

Proof. Applying Proposition 21 with M = μ_{n} and ℓ = 1 yields:

H^{1}(F, μ_{n}(α)) = \{x · K^{×n} ∈ K^{×}/K^{×n}: x\overline{x} ∈ F^{×n}\}.

If x ∈ K^{×} is such that x\overline{x} = λ^{n} with λ ∈ F^{×}, then

x′ = x(λ^{(n−1)/2}x^{−1})^{n}

represents the same element of K^{×}/K^{×n} and satisfies: x′\overline{x′} = 1. This proves the first claim.

For ℓ = 2, Proposition 21 yields a split exact sequence:

1 → H^{2}(F, μ_{n}(α)) → _{n}Br(K) ⟶^{cor} _{n}Br(F) → 1.

The proposition follows, since by the theorem of Albert–Riehm– Scharlau [16] a central simple K-algebra admits an involution of the second kind leaving F elementwise invariant if and only if its corestriction is trivial.

For a ∈ K^{×} with a\overline{a} = 1, we denote the class of a in H^{1}(F, μ_{n}(α)) by [a] and we define a commutative étale F -algebra F_{a} of dimension n as follows: on the K-algebra K(y) where y is an indeterminate subject to y^{n} = a, define an F -automorphism θ extending on K by letting θ(y) = y^{−1}. Then F_{a} is the F -subalgebra K(y)^{θ} of invariant elements under θ. It is easily seen that F_{a} ≃ F_{a′} if [a] = [a′] ∈ H^{1}(F, μ_{n}(α)). For [a] ∈ H^{1}(F, μ_{n}) = F^{×}/F^{×n}, a ∈ F^{×}, we also denote by F_{a} the F -algebra F (y) where y^{n} = a.

We now relate by cup-product the special cases Z_{n} and μ_{n} considered above. There is a canonical isomorphism: Z_{n} ⊗_{ℤ} μ_{n} = μ_{n} defined by: r^{i} ⊗ ζ ↦ ζ^{i}. Therefore, for δ_{1}, δ_{2} ∈ F^{×},

Z_{n}(δ_{1}) ⊗_{ℤ} μ_{n}(δ_{2}) = Z_{n} ⊗ μ_{n} ⊗_{ℤ}(δ_{1}) ⊗_{ℤ}(δ_{2}) = μ_{n}(δ_{1}δ_{2}).

Thus the cup-product defines a map:

H^{1}(F, Z_{n}(δ_{1})) × H^{1}(F, μ_{n}(δ_{2})) → H^{2}(F, μ_{n}(δ_{1}δ_{2})).

Proposition 26. Let K = F (\sqrt{α}) be a quadratic field extension of F and let δ_{1}, δ_{2} ∈ F^{×} be such that δ_{1}δ_{2} ≡ α mod F^{×}^{2}. Let χ ∈ H^{1}(F, Z_{n}(δ_{1})) and a ∈ F (\sqrt{δ_{2}})^{×} such that n_{F (\sqrt{α_{2}})/F} (a) = 1, with cohomology class [a] ∈ H^{1}(F, μ_{n}(δ_{2})). There exists a central simple K-algebra B(χ, a) of degree n such that

(1) [B(χ, a)] = χ ∪ [a] ∈ H^{2}(F, μ_{n}(α));

(2) B(χ, a) admits an involution σ of the second kind such that (B(χ, a), σ)_{+} contains subalgebras isomorphic to F_{χ} and to F_{a}.

Proof. We first consider the special cases where δ_{1} ∈ F^{×}^{2} or δ_{2} ∈ F^{×}^{2}. If δ_{1} ∈ F^{×}^{2}, we have χ ∈ H^{1}(F, Z_{n}) and [a] ∈ H^{1}(F, μ_{n}(α)) with a ∈ K^{×} such that a\overline{a} = 1. The algebra F_{χ} is a Galois Z_{n}-algebra over F. The central simple algebra B(χ, a) is the crossed product:

B(χ, a) = ⨁_{i=0}^{n−1}(F_{χ} ⊗_{F} K)z^{i},

where the indeterminate z is subject to the relations:

z(ℓ ⊗ k) = [(r ∗ ℓ) ⊗ k]z for ℓ ∈ F_{χ}, k ∈ K, and z^{n} = a.

An involution of the second kind σ on B(χ, a) is defined by:

σ(ℓ ⊗ k) = ℓ ⊗ \overline{k} for ℓ ∈ F_{χ}, k ∈ K, and σ(z) = z^{−1}.

Clearly, (B, σ)_{+} contains F_{χ} and the subalgebra F_{a} ⊂ K(z).

If δ_{2} ∈ F^{×}^{2}, then χ ∈ H^{1}(F, Z_{n}(α)) and a ∈ F^{×}, [a] ∈ H^{1}(F, μ_{n}). Let L be a Galois Z_{n}-algebra over K corresponding to χ. According to Proposition 24, we may also choose a Galois D_{n}-algebra structure on L, viewed as an algebra over F. The algebra B(χ, a) is the crossed product:

B(χ, a) = ⨁_{i=0}^{n−1} L z^{i}

where the indeterminate z is subject to the relations:

zℓ = (r ∗ ℓ)z for ℓ ∈ L, and z^{n} = a.

An involution of the second kind σ on B(χ, a) is defined by:

σ(ℓ) = s ∗ ℓ for ℓ ∈ L, and σ(z) = z.

By definition, it is clear that F_{a} = F (z) ⊂ (B, σ)_{+} and that the subalgebra L_{1} ⊂ L of invariant elements under s, which is isomorphic to F_{χ}, lies in (B, σ)_{+}.

Suppose now δ_{1}, δ_{2} ∉ F^{×}^{2}. Let F′ = F (\sqrt{δ_{1}}) and K′ = K(\sqrt{δ_{1}}). Let L be a Galois Z_{n}-algebra over F′ which corresponds to χ ∈ H^{1}(F, Z_{n}(δ_{1})); according to Proposition 24, we may endow it with a Galois D_{n}-algebra structure over F. Since δ_{1}δ_{2} ≡ α mod F^{×}^{2}, we may identify F (\sqrt{δ_{2}}) with a subfield of K′. Consider the following crossed product algebra over K′:

B′ = ⨁_{i=0}^{n−1}(L ⊗_{F} K) z^{i}

where z is subject to:

z(ℓ ⊗ k) = [(r ∗ ℓ) ⊗ k]z for ℓ ∈ L, k ∈ K and z^{n} = a.

This algebra represents the restriction res(χ ∪ [a]) ∈ H^{1}(F′, μ_{n}(α)). We define an involution σ′ on B′ by:

σ′(ℓ ⊗ k) = (s ∗ ℓ) ⊗ \overline{k} for ℓ ∈ L, k ∈ K, and σ′(z) = z.

On the other hand, we define a K-algebra automorphism θ on B′ by:

θ(ℓ ⊗ k) = (s ∗ ℓ) ⊗ k for ℓ ∈ L, k ∈ K, and θ(z) = z^{−1}.

The restriction of θ to the center K′ is the non-trivial automorphism over K, and θ^{2} = I; therefore, the subalgebra B = B′^{θ} of invariant elements under θ is a central simple K-algebra such that

B′ = B ⊗_{K} K′.

Since the restriction map res : H^{2}(F, μ_{n}(α)) → H^{2}(F′, μ_{n}(α)) is injective (since n is odd), it follows that B represents the cup product χ∪[a] ∈ H^{2}(F, μ_{n}(α)) and we define B(χ, a) = B. Moreover, θ and σ′ commute, hence the restriction of σ′ to B(χ, a) defines an involution σ of the second kind. The subalgebra L_{1} of L elementwise invariant under s and the subalgebra K′(z)^{θ} invariant under θ both lie in (B(χ, a), σ)_{+}; the involution σ thus satisfies the required properties, since L_{1} ≃ F_{χ} and K′(z)^{θ} ≃ F_{a}.

Proposition 27. Let B be a central simple K-algebra of degree n (odd) which admits involutions of the second kind leaving F elementwise invariant, and let δ ∈ F^{×}. If B contains a subfield isomorphic to F_{χ} for some χ ∈ H^{1}(F, Z_{n}(δ)), then B ≃ B(χ, a) for some a ∈ F (\sqrt{αδ})^{×} such that n_{F (\sqrt{αδ})/F} (a) = 1.

Proof. Suppose first δ ∈ F^{×}^{2}. Then F_{χ} ⊗ K is a maximal subfield of B which is cyclic over K, hence B is a cyclic algebra:

[B] = res_{K/F} (χ) ∪ [b] ∈ H^{2}(K, μ_{n})  (12)

for some b ∈ K^{×} (so that [b] ∈ H^{1}(K, μ_{n})), where [B] denotes the image of B in H^{2}(K, μ_{n}). Since B admits involutions of the second kind over F, we have cor_{K/F} (B) = 0; therefore,

χ ∪ cor_{K/F} ([b]) = 0 in H^{2}(F, μ_{n}).  (13)

Since n is odd, we may find m ∈ _{ℤ} such that 2m ≡ 1 mod n. Then

b ≡ b^{2m} = (b\overline{b}^{−1})^{m}n_{K/F} (b)^{m} mod K^{×}^{n};

therefore, letting a = (b\overline{b}^{−1})^{m}, we have [a] ∈ H^{1}(F, μ_{n}(α)) and

[b] = [a] + m res_{K/F} ∘ cor_{K/F} ([b]) in H^{1}(K, μ_{n}).

Substituting in equation (12), we get:

[B] = res_{K/F} (χ) ∪ [a] + m res_{K/F} (χ ∪ cor_{K/F} ([b])),

hence B ≃ B(χ, a), in view of (13). Suppose next δ ∉ F^{×}^{2} and let F′ = F (\sqrt{δ}). Since δ ∈ F′^{×}^{2}, the case already considered yields:

[B ⊗ F′] = res_{F′/F} (χ) ∪ [a′]

for some [a′] ∈ H^{1}(F′, μ_{n}(αδ)). Applying the corestriction map to both sides of this equation, we get by the projection formula:

2[B] = χ ∪ cor_{F′/F} ([a′]).

Therefore, if 2m ≡ 1 mod n,

[B] = m χ ∪ cor_{F′/F} ([a′]),

hence B ≃ B(χ, a) for [a] = m cor_{F′/F} ([a′]) ∈ H^{1}(F, μ_{n}(αδ)).

In a different direction, we have the following result on the Galois cohomology of the twisted modules μ_{n}(α):

Proposition 28. Let δ_{1}, δ_{2} ∈ F^{×}. In H^{2}(F, μ_{n}^{⊗2}(δ_{1}δ_{2})), we have:

H^{1}(F, μ_{n}(δ_{1})) ∪ H^{1}(F, μ_{n}(δ_{2})) ⊂ H^{1}(F, μ_{n}) ∪ H^{1}(F, μ_{n}(δ_{1}δ_{2})),

where the left-hand side is

{x ∪ y: x ∈ H^{1}(F, μ_{n}(δ_{1})), y ∈ H^{1}(F, μ_{n}(δ_{2}))}.

Proof. Assume first that δ_{1}, δ_{2}, δ_{1}δ_{2} ∉ F^{×2}. For i = 1, 2, let F_{i} = F (\sqrt{δ_{i}}) ⊂ F_{s} and let a_{i} ∈ F_{i}^{×} be such that n_{F_{i}/F} (a_{i}) = 1, so that [a_{i}] ∈ H^{1}(F, μ_{n}(δ_{i})). If a_{i} ∈ F^{×n}, then [a_{i}] = 0, hence [a_{1}] ∪ [a_{2}] = 0 ∈ H^{1}(F, μ_{n}) ∪ H^{1}(F, μ_{n}(δ_{1}δ_{2})). For the rest of the proof, we may thus assume a_{i} ≠ ±1 for i = 1, 2. Let

u = \frac{a_{1} − 1}{a_{1} + 1} ∈ F_{1}^{×} and v = \frac{a_{2} + 1}{a_{2} − 1} ∈ F_{2}^{×},

so that

a_{1} = \frac{1 + u}{1 − u} and a_{2} = \frac{v + 1}{v − 1}.

Since n_{F_{i}/F} (a_{i}) = 1, it follows that u and v have trace zero. Therefore, u^{2}, v^{2} ∈ F^{×} and uv^{−1} ∈ F (\sqrt{δ_{1}δ_{2}})^{×}. Let

[b] = \frac{1 − v^{2}}{1 − u^{2}} · F^{×n} ∈ H^{1}(F, μ_{n})

and

[c] = \frac{1 + uv^{−1}}{1 − uv^{−1}} · F (\sqrt{δ_{1}δ_{2}})^{×n} ∈ H^{1}(F, μ_{n}(δ_{1}δ_{2})).

We claim that

[a_{1}] ∪ [a_{2}] = [b] ∪ [c] in H^{2}(F, μ_{n}^{⊗2}(δ_{1}δ_{2})).

Let M = F (\sqrt{δ_{1}}, \sqrt{δ_{2}}) ⊂ F_{s}. Since the degree of the extension M/F is relatively prime to n, the restriction map

res_{M/F} : H^{2}(F, μ_{n}^{⊗2}(δ_{1}δ_{2})) → H^{2}(M, μ_{n}^{⊗2}(δ_{1}δ_{2})) = H^{2}(M, μ_{n}^{⊗2})

is injective; therefore, it suffices to prove the claim over M. The identity

\{\frac{1 + u}{1 − u}; \frac{v + 1}{v − 1}\} = \{\frac{1 − v^{2}}{1 − u^{2}} ; \frac{1 + uv^{−1}}{1 − uv^{−1}} \} .

holds for symbols {a, b} in Milnor’s K_{2}, a proof is given in the following Lemma 29.

Applying the norm residue homomorphism K_{2}M → H^{2}(M, μ_{n}^{⊗2}) to both sides, we get

[a_{1}] ∪ [a_{2}] = [b] ∪ [c] in H^{2}(M, μ_{n}^{⊗2}) = H^{2}(M, μ_{n}^{⊗2}(δ_{1}δ_{2})),

completing the proof in the case where δ_{1}, δ_{2}, δ_{1}δ_{2} ∉ F^{×}^{2}. If δ_{1} or δ_{2} ∈ F^{×}^{2}, the proposition is obvious. If δ_{1}δ_{2} ∈ F^{×}^{2}, we may use the same arguments as above, substituting F for F (\sqrt{δ_{1}δ_{2}}), except if u = ±v. In this case however, we have a_{1} = −a_{2}^{±1}, hence [a_{1}] = [a_{2}]^{±1} and therefore [a_{1}] ∪ [a_{2}] = 0.

Lemma 29. For u, v ≠ ±1 ∈ F^{×}, u ≠ ±v, the relation

\{\frac{1 + u}{1 − u}; \frac{v + 1}{v − 1}\} = \{\frac{1 − v^{2}}{1 − u^{2}} ; \frac{1 + uv^{−1}}{1 − uv^{−1}} \}

holds in K_{2}F.

Proof. We have {1 − u; 1 + uv^{−1}} + {1 − u; v} = {1 − u; u + v}. On the other hand the basic relation {a; b} = {a; b − a} + {a − b; b} in K_{2} (see for example [9, p. 75]) implies that

{1 − u; u + v} = {1 − u; 1 + v} − {−u − v; 1 + v}
= {1 − u; 1 + v} + {1 + v; −u − v}.

Subtracting 0 = {1 + v; −v} we get

{1 − u; 1 + uv^{−1}} − {1 + v; 1 + uv^{−1}} = {1 − u; 1 + v} − {1 − u; v}.

Replacing u, v by ±u, ±v, we get four expressions which adds to the wanted relation. Another proof is to check the formula in the function field F (u, v) by using the exact sequence of Milnor for K_{2}F (t) (see [10]) to show that

\{\frac{1 + u}{1 − u}; \frac{v + 1}{v − 1}\} − \{\frac{1 − v^{2}}{1 − u^{2}} ; \frac{1 + uv^{−1}}{1 − uv^{−1}} \}

lies in K_{2}F (u), K_{2}F and then to show that it is 0 by specialization at some places.

Remark. The Galois module μ_{n} plays a special rôle in the preceding proposition. Suppose that δ_{1}, δ_{2}, ε_{1}, ε_{2} ∈ F^{×} are such that

μ_{n}(δ_{1}) ⊗ μ_{n}(δ_{2}) = μ_{n}(ε_{1}) ⊗ μ_{n}(ε_{2}),

and that the sets {μ_{n}(δ_{1}), μ_{n}(δ_{2})} and {μ_{n}(ε_{1}), μ_{n}(ε_{2})} are disjoint (up to isomorphism). If μ_{n}(ε_{1}) and μ_{n}(ε_{2}) are different from μ_{n}, then there is an extension F̃ of F such that

H^{1}(F̃, μ_{n}(δ_{1})) ∪ H^{1}(F̃, μ_{n}(δ_{2})) ⊄ H^{1}(F̃, μ_{n}(ε_{1})) ∪ H^{1}(F̃, μ_{n}(ε_{2}))

Indeed, after a biquadratic extension of F we may assume that δ_{1}, δ_{2} ∈ F^{×}^{2}. Let then F̃ be the iterated powerseries extension F ((s))((t)). Then

s · F̃^{×n} ∪ t · F̃^{×n} ∈ H^{1}(F̃, μ_{n}) ∪ H^{1}(F̃, μ_{n})

has non-trivial residue. On the other hand,

∂(H^{1}(F̃, μ_{n}(ε))) ⊂ H^{0}(F, Z_{n}(ε)) = 0

for nontrivial ε, hence s·F̃^{×n}∪t·F̃^{×n} cannot be a sum of cup-products of elements of H^{1}(F̃, μ_{n}(ε_{1})) and H^{1}(F̃, μ_{n}(ε_{2})).

Corollary 30. Let n be an odd integer and let F be a field of characteristic relatively prime to n. Assume that μ_{n} = Z_{n}(ε) for some ε ∈ F^{×}; then every central simple F -algebra split by a Galois extension of degree 2n with dihedral Galois group is cyclic.

Proof. Let A be a central simple F -algebra split by a Galois extension L/F with dihedral Galois group of order 2n. The index of A then divides 2n. Since n is odd, we may decompose the Brauer class of A:

[A] = α_{1} + α_{2}

where α_{1} has index 1 or 2 and α_{2} has index dividing n. The element α_{1} is split by an cyclic extension C_{1}/F of degree 1 or 2. If we show that 2[A] is split by a cyclic extension C_{2}/F of degree dividing n, then α_{2} is also split by C_{2}, hence A is split by the cyclic extension C_{1} ⊗ C_{2}/F. Therefore, it suffices to show that A^{⊗2} is cyclic. Let δ ∈ F^{×} be a representative of the discriminant of L. Proposition 23 shows that L is a cyclic extension of F (\sqrt{δ}). Choosing a Galois Z_{n}- algebra structure of L over F (\sqrt{δ}), we may associate to L an element χ ∈ H^{1}(F, Z_{n}(δ)) by Proposition 24. Since A ⊗ F (\sqrt{δ}) is split by L, we have

[A ⊗_{F} F (\sqrt{δ})] = res_{F (\sqrt{δ})/F} (χ) ∪ [a] in H^{2}(F (\sqrt{δ}), μ_{n})

for some [a] ∈ H^{1}(F (\sqrt{δ}), μ_{n}) = H^{1}(F (\sqrt{δ}), μ_{n}(δ)). Taking the corestriction of both sides, we get

2[A] = χ ∪ [a′] in H^{2}(F, μ_{n}),  (14)

where [a′] = cor_{F (\sqrt{δ})/F} ([a]) ∈ H^{1}(F, μ_{n}(δ)). Since μ_{n} = Z_{n}(ε), we have Z_{n}(δ) = μ_{n}(εδ), hence relation (14) yields:

2[A] ∈ H^{1}(F, μ_{n}(εδ)) ∪ H^{1}(F, μ_{n}(δ)).

From Proposition 28, it follows that

2[A] = [b] ∪ [c]

for some [b] ∈ H^{1}(F, μ_{n}) and some [c] ∈ H^{1}(F, μ_{n}(ε)) = H^{1}(F, Z_{n}). The element [c] then defines a cyclic extension of F which splits A^{⊗2}.

We conclude by discussing the special features of the case where n = 3 (assuming that the characteristic of F is not 2 nor 3). Every cubic field extension L of F is of the form F_{χ} and also of the form F_{a}: more precisely, if the discriminant of L/F is represented by α ∈ F^{×}, then the field L ⊗_{F} F (\sqrt{α}) is cyclic over F (\sqrt{α}), hence L = F_{χ} for any χ ∈ H^{1}(F, Z_{3}(α)) representing L⊗_{F} F (\sqrt{α}). Since μ_{3} = Z_{3}(−3), we may also view L = F_{a} for [a] = χ ∈ H^{1}(F, μ_{3}(−3α)).

Proposition 31. Let K = F (\sqrt{α}) be a quadratic field extension of F and let B be a central division K-algebra of degree 3 which admits involutions of the second kind leaving F elementwise invariant. Let also L be a cubic field extension of F with discriminant δ and let χ ∈ H^{1}(F, Z_{3}(δ)) be such that L = F_{χ}. The algebra B contains an isomorphic image of L if and only if B ≃ B(χ, a) (i.e. [B] = χ∪[a] in H^{2}(F, μ_{3}(α))) for some [a] ∈ H^{1}(F, μ_{3}(αδ)). The algebra B then contains a cubic field extension L′ = F_{a} of discriminant −3αδ. Moreover, there exists an involution σ such that L, L′ ⊂ (B, σ)_{+}.

The algebra B always contains cubic field extensions L_{∗}, L′_{∗} of discriminant −3, α respectively, and carries an involution σ_{∗} such that L_{∗}, L′_{∗} ⊂ (B, σ_{∗})_{+}.

Proof. The first part follows from Propositions 26 and 27. In order to prove that B always contains extensions L_{∗}, L′_{∗} as stated, we use Proposition 28: let L be any cubic field extension of F contained in B; then, letting δ ∈ F^{×} denote a representative of the discriminant of L/F, we have

B ≃ B(χ, a)

for some χ ∈ H^{1}(F, Z_{3}(δ)) and some [a] ∈ H^{1}(F, μ_{3}(αδ)). Since Z_{3}(δ) = μ_{3}(−3δ), it follows that

[B] ∈ H^{1}(F, μ_{3}(−3δ)) ∪ H^{1}(F, μ_{3}(αδ)),

hence Proposition 28 shows that B ≃ B(χ_{∗}, a_{∗}) for some χ_{∗} ∈ H^{1}(F, μ_{3}) = H^{1}(F, Z_{3}(−3)) and some [a_{∗}] ∈ H^{1}(F, μ_{3}(−3α)) = H^{1}(F, Z_{3}(α)). We then let L_{∗} = F_{χ_{∗}} and L′_{∗} = F_{a_{∗}}.

Note that, according to condition (e) of Theorem 16, the involution σ_{∗} is distinguished; the proposition above thus yields another proof of the existence of distinguished involutions.

Note also that the pair (−3, α) is the only pair of discriminants that occurs for every central division F (\sqrt{α})-algebra of degree 3 which admits involutions of the second kind: given a pair (ε_{1}, ε_{2}) with ε_{1}ε_{2} ∈ (−3α) · F^{×}^{2}, the remark following Proposition 25 together with Proposition 27 shows that there is an algebra B over some extension of F which does not contain any cubic separable extension of discriminant ε_{1} or ε_{2}. For example, there are algebras B which do not contain a cyclic cubic field extension, i.e. a cubic field extension of discriminant 1, as already observed in the example at the end of section 4.

Since 3-fold Pfister forms are classified up to isometry by their Arason invariant:

Ar(⟨⟨a, b, c⟩⟩) = (a · F^{×}^{2}) ∪ (b · F^{×}^{2}) ∪ (c · F^{×}^{2}) ∈ H^{3}(F, μ_{2}),

(see [5]), Theorem 15 (together with Proposition 25) yields a classification of central simple algebras of degree 3 with involution by cohomological invariants:

Corollary 32. Triples (K, B, σ) where K = F (\sqrt{α}) is a quadratic étale F -algebra, B is a central simple K-algebra of degree 3 and σ is an involution of the second kind on B leaving F elementwise invariant, are classified over F by the three cohomological invariants:

\begin{aligned}f_{1}(K, B, σ) & = & α · F^{×}^{2} ∈ H^{1}(F, μ_{2}), \\ g_{2}(K, B, σ) & = & [B] ∈ H^{2}(F, μ_{3}(α)), \\ f_{3}(K, B, σ) & = & Ar(π(B, σ)) ∈ H^{3}(F, μ_{2}). \end{aligned}

Proof.

Indeed we have seen that f_{1} determines K, g_{2} determines B and f_{3} determines the involution σ.

Remark. Let K_{0}/F_{0} be a separable quadratic field extension and let B_{0} be a central division K_{0}-algebra of degree 3 which admits an involution of the second kind. Let F = F_{0}((t)), K = K_{0}((t)) and B = B_{0}((t)). Let σ be any involution on B, not necessarily defined over B_{0}. We claim that the second residue φ^{t}(Q_{σ}) is always trivial, i.e. Q_{σ} is induced from F_{0}. A cubic étale F -subalgebra L of B is either unramified or totally ramified, i.e. L = F [X]/(X^{3} − ut) for u ∈ F_{0}^{×}. Thus in both cases the discriminant of L is a class defined over F_{0}. On the other hand, an element λ ∈ L such that n_{L/F} (λ) is a square in F^{×} must be a unit in F_{0}[[t]]. This, together with Proposition 11, implies that φ^{t}(Q_{σ}) = 0. It follows that the invariant f_{3}(B, σ) cannot in general be arbitrary. In particular the three invariants f_{1}, g_{2} and f_{3} are not independent. A corresponding question for invariants of exceptional Jordan algebras is discussed in [18].

References

[1] A.A. Albert, Structure of Algebras, Colloquium Publ. 24, Amer. Math. Soc. Providence, R.I., 1939.

[2] A.A. Albert, On involutorial associative division algebras, Scripta Mathematica 26 (1963), 309–316.

[3] J.K. Arason; R. Elman, Nilpotence in the Witt ring, Amer. J. Math. 113 (1991), 861–875.

[4] E. Bayer-Fluckiger; H.W. Lenstra Jr., Forms in odd-degree extensions and self-dual normal bases, Amer. J. Math. 112 (1990), 359–373.

[5] R. Elman; T.Y. Lam, Pfister forms and K-theory of fields, J. Algebra 23 (1972) 181–213.

[6] D.E. Haile; M.-A. Knus, On division algebras of degree 3 with involutions, Preprint (1995).

[7] N. Jacobson, A note on hermitian forms, Bull. Amer. Math. Soc. 50 (1944), 645–648.

[8] N. Jacobson, Structure and Representations of Jordan Algebras, AMS Colloquium Publ. Vol. 39, Amer. Math. Soc., Providence, R.I., 1968.

[9] I. Kersten, Brauergruppen von Körpern, Aspekte der Mathematik, Vieweg, Braunschweig, 1990.

[10] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970), 318–344.

[11] H.P. Petersson; M.L. Racine, Springer forms and the first Tits construction of exceptional Jordan division algebras, manuscripta mathematica 45 (1984), 249–272.

[12] H.P. Petersson; M.L. Racine, The toral Tits process of Jordan algebras, Abh. Math. Sem. Hamburg 54 (1984), 251–256.

[13] H.P. Petersson; M.L. Racine, Cubic subfields of exceptional simple Jordan algebras, Proc. Amer. Math. Soc. 91 (1984), 31– 36.

[14] H.P. Petersson; M.L. Racine, Reduced models of Albert algebras, to appear.

[15] L.H. Rowen; D.J. Saltman, Dihedral algebras are cyclic, Proc. Amer. Math. Soc. 84 (1982), 162–164.

[16] W. Scharlau, Quadratic and Hermitian Forms, Grundlehren Math. Wiss. 270, Springer-Verlag, Berlin, 1985.

[17] J-P. Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics 5, Springer-Verlag, Berlin, 1964 and 1994.

[18] J-P. Serre, Cohomologie galoisienne: Progrès et Problèmes, Sém. Bourbaki (1993–1994), exposé 783.

[19] T.A. Springer, Oktaven, Jordan-Algebren und Ausnahmegruppen, Lecture Notes, Göttingen, 1963.

Darrell E. Haile Department of Mathematics Indiana University Bloomington, IN 47405 USA haile@ucs.indiana.edu

Max-Albert Knus Departement Mathematik ETH-Zentrum CH-8092 Zürich Switzerland knus@math.ethz.ch

Markus Rost NWFI-Mathematik Universität Regensburg D-93053 Regensburg Germany markus.rost@mathematik.uni-regensburg.de

Jean-Pierre Tignol Institut de Mathématique Pure et Appliquée Université catholique de Louvain B-1348 Louvain-la-Neuve Belgium tignol@agel.ucl.ac.be