JACOBSON’S THEOREM FOR BILINEAR FORMS IN CHARACTERISTIC 2

AHMED LAGHRIBI

Abstract

The aim of this note is to extend to bilinear forms in characteristic 2 a result of Jacobson which states that over any field, two Albert quadratic forms are similar if and only if they have the same Clifford invariant.

1. Introduction

Throughout this note F denotes a field. To a biquaternion algebras Q_{1} ⊗_{F} Q_{2}, we attach a 6-dimesnional quadratic form φ given by N_{Q_{1}} ⊥ −N_{Q_{2}} ≃ H ⊥ φ, where N_{Q_{i}} is the norm form of the quaternion algebra Q_{i}, and H is the hyperbolic plane (⊥ and ≅ mean the orthogonal sum and isometry, respectively). The form φ has trivial signed discriminant (resp. trivial Arf invariant) if the characteristic is ≠ 2 (resp. if the characteristic is 2). We call such a form an Albert quadratic form. A well-known result of Jacobson states that two biquaternion algebras are isomorphic if and only if their corresponding Albert quadratic forms are similar [3]. In other words, this result says that two Albert quadratic forms are similar if and only if they have the same Clifford invariant. Using a method based on quadratic forms theory, Mammone and Shapiro recovered Jacobson’s result [7], and also completed it in characteristic 2 (see [7, comments in the middle of page 529]). The Clifford invariant of a quadratic form (nonsingular quadratic form if the characteristic is 2) is defined in the 2-torsion_{2}Br(F) of the Brauer group of F. It is well-known that _{2}Br(F) is isomorphic to I^{2}F/I^{3}F (resp. I_{q}^{2}F/I_{q}^{3}F) if the characteristic is not 2 (resp. if the characteristic is 2), where I^{k}F = (IF)^{k}, I_{q}^{k}F = I^{k−1}F ⊗ W_{q}(F) and IF denotes the ideal of the Witt ring of even dimensional quadratic forms or bilinear forms according as the characteristic is different or equal to 2, and W_{q}(F) denotes the Witt group of nonsingualr quadratic forms [1]. These isomorphisms are due to Merkurjev [8] and Sah [10], respectively. Since in characteristic 2 we should distinguish between quadratic and bilinear forms, it is natural to ask whether an analogue of Jacobson’s result holds for bilinear forms in characteristic 2. As for quadratic forms, an Albert bilinear form means a 6-dimensional form with trivial determinant. Of course there is no notion of Clifford invariant for bilinear forms in characteristic 2, but we have a result of Kato which gives an analogue of Merkurjev’s and Sah’s results cited before (see Theorem 2.1), and where the group of finite sums of logarithmic differential forms (see below) is used as for _{2}Br(F) in the case of quadratic forms. We will see that this ingredient suffices to get a result similar to that of Jacobson.

Recall that for n ≥ 1, one denotes by Ω_{F}^{n} = ⋀^{n} Ω_{F}^{1} the vector space of n-differential forms over F, where Ω_{F}^{1} is the F -vector space generated by symbols dx, x ∈ F, subject to the relations: d(x + y) = dx + dy and d(xy) = xdy + ydx, for x, y ∈ F. An element \frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}} is called an n-logarithmic symbol. A sum of n-logarithmic symbols has length k if it is a sum of k n-logarithmic symbols but not a sum of less that k n-logarithmic symbols.

For a_{1}, ⋯, a_{n} ∈ F^{∗} := F \{0}, let ⟨a_{1}, ⋯, a_{n}⟩_{b} denote the bilinear form given by ∑_{i=1}^{n} a_{i}x_{i}y_{i}. It is clear that any Albert bilinear form γ is isometric to α ⟨r, s, rs, u, v, uv⟩_{b}

2000 Mathematics Subject Classification 11E04 (primary), 11E81 (secondary).

for suitable scalars α, r, s, u, v ∈ F^{∗}. We associate to such a form γ the following sum of 2-logarithmic symbols:

e^{2}(γ) := \frac{dr}{r} ∧ \frac{ds}{s} + \frac{du}{u} ∧ \frac{dv}{v} ∈ Ω_{F}^{2} .

This is an invariant of γ modulo I^{3}F (see below). Our main result in this note is the following theorem: Theorem 1.1. Let F be a field of characteristic 2, and let γ_{1}, γ_{2} be two Albert bilinear forms. Then we have the following statements:

(1) The length of e^{2}(γ_{1}) is 2, 1 or 0 according as the Witt index of γ_{1} is 0, 1 or 3.

(2) γ_{1} is similar to γ_{2} if and only if e^{2}(γ_{1}) = e^{2}(γ_{2}).

By using Theorem 2.1 of Kato, we reduce the proof of Theorem 1.1 to the use of methods from quadratic and bilinear forms theory as was used in [7]. However, some of our results differ from those given in [7], and their proofs require more details. The reason is that for bilinear forms in characteristic 2 some classical results, like the Witt cancellation and the representation criterion, fail. Moreover, we will be based on the connection between totally singular quadratic forms and bilinear forms, and the notion of norm degree introduced in [2, Section 8] will play a crucial rôle.

The rest of this note is organized as follows. We finish this section by giving backgrounds on quadratic and bilinear forms in characteristic 2. The next section in devoted to some results needed for the proof of Theorem 1.1, and more specifically it will concern the similarity of 4-dimensional bilinear forms in characteristic 2, and then in the third section we prove the theorem.

Form now on, we assume that F is of characteristic 2. The expression “bilinear form” means “regular symmetric bilinear form of finite dimension”. To keep this note self-contained we briefly recall some notions. More details can be found in [1], [2].

For a quadratic (or bilinear) form φ, we denote by dim φ its dimension. Two quadratic (or bilinear) forms φ and ψ are called similar if φ ≅ αψ for some scalar α ∈ F^{∗}. A quadratic (or bilinear) form φ is called a subform of another form ψ if there exists a form φ′ such that ψ ≅ φ ⊥ φ′.

To any bilinear form B with underlying vector space V, we associate a unique quadratic form B̃ given by: B̃(v) = B(v, v) for v ∈ V. A quadratic form φ is called totally singular if φ ≅ B̃ for some bilinear form B. If B ≅ ⟨a_{1}, ⋯, a_{n}⟩_{b}, then we denote B̃ by ⟨a_{1}, ⋯, a_{n}⟩.

For a field extension K/F and a quadratic (or bilinear) form φ, the form φ ⊗ K is denoted by φ_{K}.

For a quadratic form φ, we denote by F (φ) its function field, i.e., the function field of the affine quadric given by φ = 0. The function field of a bilinear form B is by definition the field F (B̃).

A quadratic form φ with underlying vector space V is isotropic if there exists v ∈ V \{0} such that φ(v) = 0. A bilinear form B is isotropic if B̃ is isotropic too. A form (quadratic or bilinear) is anisotropic if it is not isotropic.

Any bilinear form B decomposes as B ≅ B_{an} ⊥ M_{1} ⊥ ⋯ ⊥ M_{n}, where B_{an} is anisotropic

and M_{i} is given by the matrix (\begin{matrix}a_{i} & 1 \\ 1 & 0 \end{matrix}) for some a_{i} ∈ F^{∗} (1 ≤ i ≤ n). The form B_{an} is unique

up to isometry, we call it the anisotropic part of B. The interger n is called the Witt index of B and it is denoted by i_{W} (B). The form B is called metabolic if 2i_{W} (B) = dim B. Two bilinear forms B and C are called equivalent, denoted by B ∼ C, if B ⊥ M ≅ C ⊥ M′ for some metabolic forms M and M′. By the uniqueness of the anisotropic part the condition B ∼ C implies B_{an} ≅ C_{an}. Recall that the isotropy of a bilinear form B is equivalent to say that i_{W} (B) ≥ 1.

The form B = ⟨1, a_{1}⟩_{b} ⊗ ⋯ ⊗ ⟨1, a_{n}⟩_{b} is called an n-fold bilinear Pfister form, and is denoted by ⟨⟨a_{1}, ⋯, a_{n}⟩⟩_{b}. In this case, we denote B̃ by ⟨⟨a_{1}, ⋯, a_{n}⟩⟩ and we call it an n-fold quasi-Pfister form. Recall that a bilinear Pfister form is isotropic if and only if it is metabolic, and for any integer n ≥ 1, the ideal I^{n}F is additively generated by n-fold bilinear Pfister forms.

We say that a totally singular form φ is a quasi-Pfister neighbor if there exists a quasi-Pfister form π such that φ is similar to a subform of π and 2 dim φ > dim π. In this case, π is unique up to isometry, and for any field extension K/F, the forms φ_{K} and π_{K} are simultaneously isotropic or anisotropic.

The norm field of a nonzero totally singular form φ, denoted by N_{F} (φ), is the field F^{2}(αβ | α, β ∈ D_{F} (φ)), where D_{F} (φ) is the set of scalars in F^{∗} represented by φ. We denote by ndeg_{F} (φ) the integer [N_{F} (φ) : F^{2}] and we call it the norm degree of φ. It is clear that N_{F} (φ) = N_{F} (αφ) for any scalar α ∈ F^{∗}. If φ is anisotropic and 2^{n} < dim φ ≤ 2^{n+1}, then ndeg_{F} (φ) ≥ 2^{n+1}, and ndeg_{F} (φ) = 2^{n+1} if and only if φ is a quasi-Pfister neighbor. If ψ is a quadratic form such that φ_{F (ψ)} is isotropic, then ψ is totally singular and N_{F} (ψ) ⊂ N_{F} (φ). Moreover, there is a bijection between anisotropic n-fold quasi-Pfister forms and purely inseparable extensions of F^{2} of degree 2^{n} inside F, it is given by F^{2}(a_{1}, ⋯, a_{n}) ↔ ⟨⟨a_{1}, ⋯, a_{n}⟩⟩. We refer to [2, Section 8] for more details on norm field and some of its applications.

2. Preliminaries

It is clear that the map d : F ⟶ Ω_{F}^{1} : x ↦ dx, extends to a map d : Ω_{F}^{n} ⟶ Ω_{F}^{n+1} defined by:

d (xdx_{1} ∧ dx_{2} ∧ ⋯ ∧ dx_{n}) = dx ∧ dx_{1} ∧ dx_{2} ∧ ⋯ ∧ dx_{n}.

There is a well-defined homomorphism ℘_{n} : Ω_{F}^{n} ⟶ Ω_{F}^{n} /dΩ_{F}^{n−1} given on generators by:

℘_{n} (x\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}} ) = \overline{(x^{2} − x)\frac{dx_{1}}{x_{1}} ∧ ⋯ ∧ \frac{dx_{n}}{x_{n}}} .

We write ν_{F} (n) the kernel of this map. A crucial result that we will use is the following theorem due to Kato which gives a link between this kernel and bilinear forms. Kato also established a relation between the cokernel of ℘_{n} and quadratic forms, but we don’t need it here. Theorem 2.1 ([4]). For any integer n ≥ 1, there is an isomorphism e^{n} : I^{n}F/I^{n+1}F ⟶^{∼} ν_{F} (n), given by:

e^{n} (∑ ⟨⟨a_{1}, ⋯, a_{n}⟩⟩_{b} + I^{n+1}F ) = ∑ \frac{da_{1}}{a_{1}} ∧ ⋯ ∧ \frac{da_{n}}{a_{n}} .

We will use this theorem in the case n = 2. Another result that we need is the following theorem which gives information on the dimensions of bilinear forms in I^{n}F:

Theorem 2.2. Let B ∈ I^{n}F be anisotropic (n ≥ 1). Then:

(1) ([6]) dim B = 0 or dim B ≥ 2^{n}.

(2) If dim B > 2^{n}, then dim B ≥ 2^{n} + 2^{n−1}.

Statement (2) can be deduced from a result of Vishik [11] by the same argument used for the proof of [6, Prop. 5.7].

Lemma 2.3. Let B be an anisotropic bilinear form and d ∈ F^{∗}\F^{∗2}. Then, B becomes metabolic over F (\sqrt{d}) if and only if B ≅ α_{1} ⟨1, d + x_{1}^{2}⟩_{b} ⊥ ⋯ ⊥ α_{n} ⟨1, d + x_{n}^{2}⟩_{b} for some α_{i}, x_{i} ∈ F with α_{i} ≠ 0, 1 ≤ i ≤ n.

Proof. Use [5, Lem. 3.4] and the fact that for a ≠ 0, b ∈ F, the bilinear form given by the

matrix (\begin{matrix}a & b \\ b & ad \end{matrix}) is isometric to α ⟨1, d + x^{2}⟩_{b} for some α ≠ 0, x ∈ F.

Some results on bilinear forms of dimension 4 will be needed, more particularly properties on the similarity between such forms. Before we give our contribution in this direction (Proposition 2.5 and Corollary 2.6), we start by clarifying the situation whether a 4-dimensional bilinear form becomes isotropic over the inseparable quadratic extension given by its determinant. Recall that an anisotropic quadratic form of dimension 4 (nonsingular if the characteristic is 2) stays anisotropic over the quadratic extension given by its signed discriminant (or the separable quadratic extension given by its Arf invariant). For bilinear forms in characteristic 2 the situation is different as shows the following proposition:

Proposition 2.4. Let B be a bilinear form of dimension 4 whose determinant is not trivial. Then, B becomes isotropic over the quadratic inseparable extension given by its determinant if and only if ndeg_{F} (B̃) ≤ 4.

Proof. Write B = α ⟨r, s, rs, d⟩_{b} for suitable scalars α, r, s ∈ F^{∗}. One has N_{F} (B̃) = F^{2}(r, s, d) and thus ndeg_{F} (B̃) ≤ 8. We may suppose that B is anisotropic, in particular ⟨r, s, rs⟩_{b} is anisotropic too, and thus [F^{2}(r, s) : F^{2}] = 4. Hence, ndeg_{F} (B̃) ∈ {4, 8}.

Suppose ndeg_{F} (B̃) = 4. Hence, d ∈ F^{2}(r, s). Since d ∉ F^{2}, one can write F^{2}(r, s) = F^{2}(d, k) for some k ∈ F^{∗}. In particular, ⟨⟨r, s⟩⟩ ≅ ⟨⟨d, k⟩⟩. Now it is clear that ⟨⟨r, s⟩⟩_{F (\sqrt{d})} is isotropic. Then, ⟨r, s, rs, d⟩_{F (\sqrt{d})} is also isotropic, i.e., B_{F (\sqrt{d})} is isotropic.

Conversely, if B_{F (\sqrt{d})} is isotropic, then (⟨⟨r, s⟩⟩_{b})_{F (\sqrt{d})} is metabolic. By Lemma 2.3 it is clear ⟨⟨r, s⟩⟩_{b} ≅ ⟨⟨d + x^{2}, y⟩⟩ for suitable scalars x, y ∈ F. Hence, d ∈ F^{2}(r, s) and then ndeg_{F} (B̃) = 4.

The following proposition is in the spirit of a result due to Wadsworth [12, Theorem 7]. In our case, the notion of norm degree plays an essential rôle:

Proposition 2.5. Let B = ⟨r, s, rs, d⟩_{b} and C = ⟨u, v, uv, d⟩_{b} be two anisotropic bilinear forms of dimension 4 having the same determinant d. Suppose that ndeg_{F} (B̃) = 8 and ⟨r, s, rs, u, v, uv⟩ is isotropic. Then, B and C are similar over F (\sqrt{d}) if and only if there exists x ∈ F^{∗} such that ⟨r, s, rs, d + x^{2}⟩ is similar to ⟨u, v, uv, d + x^{2}⟩.

Proof. It is clear that N_{F} (B̃) = F^{2}(r, s, d). Suppose that ⟨r, s, rs, d + x^{2}⟩_{b} is similar to ⟨u, v, uv, d + x^{2}⟩_{b} for some x ∈ F. Since (⟨d + x^{2}⟩_{b})_{F (\sqrt{d})} ≅ (⟨1⟩_{b})_{F (\sqrt{d})} ≅ (⟨d⟩_{b})_{F (\sqrt{d})}, it is clear that the forms ⟨r, s, rs, d⟩_{b} and ⟨u, v, uv, d⟩_{b} are similar over F (\sqrt{d}).

Conversely, suppose that B_{F (\sqrt{d})} is similar to C_{F (\sqrt{d})}. Then, by using the multiplicativity of bilinear Pfister forms, we get that (⟨⟨r, s⟩⟩_{b} ⊥ ⟨⟨u, v⟩⟩_{b})_{F (\sqrt{d})} is metabolic. The isotropy of ⟨r, s, rs, u, v, uv⟩_{b} and Lemma 2.3 imply that

⟨⟨r, s⟩⟩_{b} ⊥ ⟨⟨u, v⟩⟩_{b} ∼ α ⟨1, d + x^{2}⟩_{b} ⊥ β ⟨1, d + y^{2}⟩_{b}

for suitable α, β, x, y ∈ F with α, β ≠ 0. But by comparing determinants in the last relation, we may suppose that x = y. Moreover, by the subform theorem for bilinear forms [5, Prop. 1.1], we may suppose that α is represented by ⟨r, s, rs⟩_{b}, and thus ⟨⟨r, s⟩⟩_{b} ≅ α ⟨⟨r, s⟩⟩_{b}. Hence we get

α(⟨⟨r, s⟩⟩_{b} ⊥ ⟨1, d + x^{2}⟩_{b}) ∼ ⟨⟨u, v⟩⟩_{b} ⊥ β ⟨1, d + x^{2}⟩_{b} .

The anisotropic part of ⟨⟨r, s⟩⟩_{b} ⊥ ⟨1, d + x^{2}⟩_{b} has dimension 4, and thus it is isometric to ⟨r, s, rs, d + x^{2}⟩_{b}, otherwise the form ⟨r, s, rs⟩_{b} would represent d + x^{2}, and thus d ∈ F^{2}(r, s), a contradiction with ndeg_{F} (B̃) = 8. By the uniqueness of the anisotropic part, the form ⟨⟨u, v⟩⟩_{b} ⊥ β ⟨1, d + x^{2}⟩_{b} also has an anisotropic part of dimension 4. It follows from the multiplicativity of bilinear Pfister forms that α ⟨r, s, rs, d + x^{2}⟩_{b} ≅ λ ⟨u, v, uv, d + x^{2}⟩_{b} for some scalar λ ∈ F^{∗}. Hence, the claim.

We don’t know if Proposition 2.5 remains true in the case of norm degree 4. As a corollary of this proposition we get the following:

Corollary 2.6. Let B and C be anisotropic bilinear forms of dimension 4 such that B ⊥ C ∈ I^{3}F. Then, B is similar to C.

Proof. The forms B and C have the same determinant since B ⊥ C ∈ I^{3}F. Set B = α ⟨r, s, rs, l⟩_{b} and C = β ⟨u, v, uv, l⟩_{b}. We may suppose that l is not a square, otherwise we get the similarity by Theorem 2.2 and the multiplicativity of bilinear Pfister forms.

Since B ⊥ C ∈ I^{3}F, it follows from Theorem 2.1 that e^{2}(B ⊥ C) = \frac{du}{u} ∧ \frac{dv}{v} + \frac{dr}{r} ∧ \frac{ds}{s} + \frac{d(αβ)}{αβ} ∧ \frac{dl}{l} = 0. Hence, by statement (1) of Theorem 1.1 the form ⟨r, s, rs, u, v, uv⟩_{b} is isotropic (note that we can use statement (1) of Theorem 1.1 since its proof is independent of this corollary). Moreover, Theorem 2.2(1) implies that the forms B_{F (C)} and C_{F (B)} are isotropic, and thus N_{F} (B̃) = N_{F} (C̃) = F^{2}(r, s, l). Since B ⊥ C ∈ I^{3}F, we get (⟨⟨r, s⟩⟩_{b} ⊥ ⟨⟨u, v⟩⟩_{b})_{F (\sqrt{l})} ∈ I^{3}F (\sqrt{l}), and again by Theorem 2.2(1) we deduce the isometry (⟨⟨r, s⟩⟩_{b})_{F (\sqrt{l})} ≅ (⟨⟨u, v⟩⟩_{b})_{F (\sqrt{l})}. Now it is clear that B_{F (\sqrt{l})} similar to C_{F (\sqrt{l})}. We discuss two cases:

(1) Suppose that ndeg_{F} (B̃) = 8: In this case we conclude by Proposition 2.5 that ⟨r, s, rs, l + x^{2}⟩_{b} ≅ k ⟨u, v, uv, l + x^{2}⟩_{b} for some scalars x, k ≠ 0 ∈ F. In particular,

⟨⟨r, s⟩⟩_{b} ∼ ⟨1, l + x^{2}⟩_{b} ⊥ k ⟨⟨u, v⟩⟩_{b} ⊥ k ⟨1, l + x^{2}⟩_{b} .

If we combine this relation with B ⊥ C ∈ I^{3}F, it is clear that modulo I^{3}F we get

αk ⟨1, l + x^{2}⟩_{b} ⊥ α ⟨l, l + x^{2}⟩_{b} ⊥ β ⟨1, l⟩_{b} ∈ I^{3}F.

Hence

αk ⟨1, l + x^{2}⟩_{b} ⊥ α ⟨l, l + x^{2}⟩_{b} ∼ β ⟨1, l⟩_{b} .

Since ⟨r, s, rs, l⟩_{b} ∼ k ⟨⟨u, v⟩⟩_{b} ⊥ k ⟨1, l + x^{2}⟩_{b} ⊥ ⟨l, l + x^{2}⟩_{b}, it follows that ⟨r, s, rs, l⟩_{b} ∼ k ⟨⟨u, v⟩⟩_{b} ⊥ αβ ⟨1, l⟩_{b}. By the uniqueness of the anisotropic part and the multiplicativity of bilinear Pfister forms, we conclude that B ≅ mC for some scalar m ∈ F^{∗}.

(2) Suppose that ndeg_{F} (B̃) = 4: Since ⟨r, s, rs⟩_{b} is anisotropic, the form ⟨⟨r, s⟩⟩_{b} is anisotropic too, and then [F^{2}(r, s) : F^{2}] = 4. Hence, N_{F} (B̃) = N_{F} (C̃) = F^{2}(r, s) = F^{2}(u, v) and l ∈ F^{2}(r, s).

(i) If B ⊥ C is isotropic, then it is metabolic and thus B ≅ C.

(ii) If B ⊥ C is anisotropic. By Theorem 2.2(1), B ⊥ C becomes metabolic over its function field. It follows from [5, Cor. 5.5] that B ⊥ C is similar to 3-fold bilinear Pfister form. Hence, ndeg_{F} ({ \atop B ⊥ C}) = 8 = ndeg_{F} (α({ \atop B ⊥ C})). By a simple computation with the fact l ∈ F^{2}(r, s) =

F^{2}(u, v), one has N_{F} (α({ \atop B ⊥ C})) = F^{2}(r, s, αβ). Moreover, ⟨⟨r, s⟩⟩ ≅ ⟨⟨u, v⟩⟩ since F^{2}(r, s) =

F^{2}(u, v). Hence, ⟨u, v, uv⟩ is isotropic over K := F (⟨⟨r, s⟩⟩) since it is a quasi-Pfister neighbor of ⟨⟨u, v⟩⟩. Consequently, (⟨u, v, uv⟩_{b})_{K} ∼ (⟨1⟩_{b})_{K} and similarly (⟨r, s, rs⟩_{b})_{K} ∼ (⟨1⟩_{b})_{K}. Hence, after extending the relation α(B ⊥ C) ∈ I^{3}F to the field K, we conclude that (⟨⟨l, αβ⟩⟩_{b})_{K} is metabolic. In particular, ⟨⟨l, αβ⟩⟩ ≅ ⟨⟨e, f⟩⟩ for some e, f ∈ F^{2}(r, s) [5, Th. 1.2]. This implies that αβ ∈ F^{2}(r, s), and thus [F^{2}(r, s, αβ) : F^{2}] = 4, a contradiction.

3. Proof of Theorem 1.1

(1) (i) If e^{2}(γ_{1}) has length 2, then it is clear that i_{W} (γ_{1}) = 0.

(ii) If e^{2}(γ_{1}) has length 1: Let k, l ∈ F^{∗} be such that e^{2}(γ_{1}) = \frac{dk}{k} ∧ \frac{dl}{l}. Let K be the function field of τ = ⟨⟨k, l⟩⟩_{b}. Since τ_{K} is metabolic, one has e^{2}(γ_{1})_{K} = 0. It follows from Theorem 2.1 that (γ_{1})_{K} ∈ I^{3}K, and by Theorem 2.2 (γ_{1})_{K} is metabolic. Since ndeg_{F} (τ̃) = 4 because τ is anisotropic, it follows from [5, Th. 1.2] that γ_{1} is isotropic. Moreover, the form γ_{1} is not metabolic by reason of lenght, hence i_{W} (γ_{1}) = 1.

(iii) If e^{2}(γ_{1}) has length 0, then it follows from Theorems 2.1 and 2.2 that i_{W} (γ_{1}) = 3.

(2) Let γ_{1}, γ_{2} be two Albert bilinear forms. We have to show that γ_{1} is similar to γ_{2} if and only if e^{2}(γ_{1}) = e^{2}(γ_{2}).

Suppose that γ_{1} is similar to γ_{2}. Then, γ_{1} ⊥ γ_{2} ∈ I^{3}F. Il follows from Theorem 2.1 that e^{2}(γ_{1}) = e^{2}(γ_{2}).

Conversely, suppose that e^{2}(γ_{1}) = e^{2}(γ_{2}). Then, again by Theorem 2.1 γ_{1} ⊥ γ_{2} ∈ I^{3}F. After multiplying, if necessary, γ_{1} and γ_{2} by suitable scalars, we may suppose that γ_{1} ⊥ γ_{2} is isotropic. By Theorem 2.2(2) the Witt index of γ_{1} ⊥ γ_{2} is at least 2. Let k, l ∈ F^{∗} be such that γ_{1} ≅ k ⟨1, l⟩ ⊥ B and γ_{2} = k ⟨1, l⟩ ⊥ C for some 4-dimensional bilinear forms B and C which have the same determinant l. Write B = α ⟨r, s, rs, l⟩ and C = β ⟨u, v, uv, l⟩. An easy computation of e^{2}(γ_{1}) = e^{2}(γ_{2}) gives that

\frac{dr}{r} ∧ \frac{ds}{s} + \frac{dl}{l} ∧ \frac{d(kα)}{kα} = \frac{du}{u} ∧ \frac{dv}{v} + \frac{dl}{l} ∧ \frac{d(kβ)}{kβ}.

In particular,

\frac{dr}{r} ∧ \frac{ds}{s} + \frac{du}{u} ∧ \frac{dv}{v} = \frac{dl}{l} ∧ \frac{d(k^{2}αβ)}{k^{2}αβ}.

We conclude by statement (1) that ⟨r, s, rs, u, v, uv⟩_{b} is isotropic. Since B ⊥ C ∈ I^{3}F, it follows from Corollary 2.6 that B ≅ mC for some scalar m ∈ F^{∗}. Since γ_{1} ⊥ γ_{2} ∈ I^{3}F, one deduces that mγ_{1} ⊥ γ_{2} ∈ I^{3}F, i.e., mk ⟨1, l⟩ ⊥ k ⟨1, l⟩ ∈ I^{3}F. Hence, mk ⟨1, l⟩ ≅ k ⟨1, l⟩, and γ_{1} ≅ mγ_{2}.

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Ahmed Laghribi Faculté Jean Perrin Laboratoire de Mathématiques de Lens Rue Jean Souvraz - SP18 F-62307 Lens

laghribi@euler.univ-artois.fr