Isotropy of quadratic forms over the function field of a quadric in characteristic 2

Detlev W. Hoffmann^{a,1} and Ahmed Laghribi^{b,1}

aLaboratoire de Mathématiques, UMR 6623 du CNRS, Université de Franche-Comté, 16 route de Gray, F-25000 Besançon Cedex, France

bUniversität Bielefeld, Fakultät für Mathematik, Postfach 100131, D-33501 Bielefeld

Abstract

We extend to characteristic 2 a theorem by the first author which states that if φ and ψ are anisotropic quadratic forms over a field F such that dim φ ≤ 2^{n} < dim ψ for some nonnegative integer n, then φ stays anisotropic over the function field F (ψ) of ψ. The case of singular forms is systematically included. We give applications to the characterization of quadratic forms with maximal splitting. We also prove a characteristic 2 version of a theorem by Izhboldin on the isotropy of φ over F (ψ) in the case dim φ = 2^{n} + 1 ≤ dim ψ.

Key words: Quadratic forms, function field of a quadratic form, Pfister forms, Pfister neighbors, dominated quadratic forms, standard splitting of a quadratic form, maximal splitting. 2000 MSC: Primary 11E04, Secondary 11E81

Table of contents

1. Introduction

2. Preliminaries

3. Domination of quadratic forms by Pfister forms and proofs of Theorems 1.1 and 1.3

  Email addresses: detlev@math.univ-fcomte.fr,
laghribi@Mathematik.Uni-Bielefeld.DE (Ahmed Laghribi).

1 Both authors have been supported in part by the European research networks FMRX-CT97-0107 and HPRN-CT-2002-00287 “Algebraic K-Theory, Linear Algebraic Groups and Related Structures”, and by the program INTAS 99-00817 “Linear Algebraic Groups and Related Linear and Homological Structures.”

4. Quadratic forms with maximal splitting

5. Proof of Theorem 1.2

1 Introduction

An important problem in the algebraic theory of quadratic forms over a field F is to classify anisotropic quadratic forms ψ for which a given anisotropic quadratic form φ becomes isotropic over F (ψ), the function field of ψ.

In characteristic different from 2, the first author proved a general result on this problem which asserts that φ stays anisotropic over F (ψ) if dim φ ≤ 2^{n} < dim ψ for some integer n ≥ 1 [9, Th. 1]. A partial generalization of this result to characteristic 2 was given by the second author and Mammone [20], more precisely it was shown that φ stays anisotropic over F (ψ) in the case dim φ+dim ql(φ) ≤ 2^{n} < dim ψ, where ql(φ) denotes the quasilinear part of φ. In particular, this gives a complete generalization in the case of nonsingular quadratic forms, i.e. quadratic forms φ with dim ql(φ) = 0. Recall that in characteristic different from 2, the result [9, Th. 1] is also a consequence of Rost’s degree formula [23], or of the very general results by Karpenko [15] on the first Witt index of quadratic forms and by Karpenko and Merkurjev on the essential dimension of quadrics [16].

The main result of this paper is the following theorem which completely extends [9, Th. 1] to characteristic 2, and thus gives a positive answer to the question asked at the end of [20]:

Theorem 1.1 Let F be a field of characteristic 2. Let φ, ψ be anisotropic quadratic forms over F (possibly singular) such that dim φ ≤ 2^{n} < dim ψ for some integer n ≥ 1. Then, φ stays anisotropic over F (ψ).

One of the consequences of Theorem 1.1 is that an anisotropic quadratic form φ of dimension 2^{n} + m with 0 < m ≤ 2^{n} satisfies i_{t}(φ_{F (φ)}) ≤ m, where i_{t}(ψ) denotes the total index of a quadratic form ψ, i.e. the maximal dimension of a totally isotropic subspace inside the underlying vector space of ψ (cf. also Definition 2.4 and Lemma 4.1). When i_{t}(φ_{F (φ)}) = m we say that φ has maximal splitting. Examples of such quadratic forms are Pfister neighbors and any quadratic form φ with dim φ = 2^{n} +1 for some integer n ≥ 0 (cf. Section 4 for more details). Our main result on quadratic forms with maximal splitting is the following theorem:

Theorem 1.2 Let F be a field of characteristic 2. Let ψ be an anisotropic not totally singular quadratic form over F such that dim ψ ≤ 2^{n+1} and such that one of the following conditions holds:

(1) dim ψ ≥ 2^{n+1} − 2 with n ≥ 2;

(2) dim ψ = 2^{n+1} − 3 with n ≥ 2 if dim ql(ψ) = 1, and n ≥ 3 otherwise;

(3) dim ψ = 2^{n+1} − 4 with n ≥ 3;

(4) dim ψ = 2^{n+1} − 5 with n ≥ 3 and (dim ql(ψ) = 1 or ql(ψ) is similar to ⟨1, a, b, ab⟩ ⊥ ⟨c⟩ for some a, b, c ∈ F^{∗});

(5) dim ψ = 2^{n+1} − 6 and ψ ∈ I^{2}W_{q}(F) with n ≥ 3.

If ψ has maximal splitting, then it is a Pfister neighbor.

We also generalize to characteristic 2 a theorem by Izhboldin [12, Th. 0.2] except in the case of totally singular quadratic forms:

Theorem 1.3 Let F be a field of characteristic 2, and let φ, ψ be anisotropic quadratic forms over F such that φ is not totally singular and dim φ = 2^{n} + 1. If dim ψ > 2^{n} and φ_{F (ψ)} is isotropic, then ψ is not totally singular and ψ_{F (φ)} is also isotropic.

Again, it should be remarked that in characteristic not 2, Izhboldin’s theorem can easily be deduced from Karpenko and Merkurjev’s results in [15], [16]. It would be very interesting to know in how far their methods and results carry over to the case of characteristic 2 and possibly singular forms.

We combine Theorems 1.2, 1.3 and Proposition 4.6 to get:

Corollary 1.4 Let F be field of characteristic 2, and let φ, ψ be anisotropic quadratic forms over F such that φ is not totally singular, dim φ = 2^{n} + 1 for some integer n ≥ 1, and ψ satisfies one of the conditions (1)–(5) of Theorem 1.2 for this integer n. If φ_{F (ψ)} is isotropic, then φ and ψ are Pfister neighbors of the same (n + 1)-fold Pfister form.

This paper is organized as follows. The next section is devoted to defining the notions and providing some preliminary results which we will need in the proofs. This includes Witt decomposition of a quadratic form, the domination relation (which is introduced since we don’t exclude singular forms), the analogue of the Cassels-Pfister subform theorem, some facts about Pfister neighbors and the standard splitting of a quadratic form. Most of this can also be found in this or similar form in earlier articles by the authors [18], [19], [10], but since many facts in characteristic 2 are not too widely known and in an attempt to keep the paper as self-contained as possible, we decided to give a fairly detailed account of the notions, facts, and preliminary results which we will use throughout the paper.

Section 3 is devoted to the proofs of Theorem 1.1 and Theorem 1.3.

Before proving Theorem 1.1 at the end of subsection 3.1, we first provide the main ingredient (Proposition 3.1) for the proof of the theorem. This proposition is based on the idea that an anisotropic quadratic form becomes dominated by a Pfister form after extending scalars to a suitable extension of the ground field. This was used originally by the first author in his proof of [9, Th. 1]. However, in our case we have to apply special care as we include singular quadratic forms without any further hypotheses on their quasilinear parts, contrary to what was made in [20], and thus some known results in characteristic different from 2, like Witt cancellation, cannot be carried over to characteristic 2. To avoid this difficulty we use a new idea from [10] called “completion lemma” (cf. subsection 2.4) which allows us to extend an isometry between singular forms to one between suitable nonsingular forms which dominate them.

The proof of 1.3 in subsection 3.2 is based on the same idea already employed by Izhboldin [12] in characteristic ≠ 2. More precisely, we will show and use the fact that an anisotropic not totally singular quadratic form φ of dimension 2^{n} + 1 will be dominated by an anisotropic (n + 1)-fold Pfister form after extending scalars to a suitable field extension E/F with the property that E(φ)/F (φ) is unirational.

Various general and rather straightforward results concerning quadratic forms with maximal splitting are given in section 4.

In section 5, we prove Theorem 1.2. This is done case by case, using an important result on Pfister neighbors (due to Knebusch in characteristic ≠ 2), and a result which extends to characteristic 2 a theorem due to Fitzgerald ([10, Th. 6.6], [10, Th. 5.1, Cor. 5.3 ], see Theorem 2.19 in the present paper), some results on division algebras like the index reduction theorem, some descent techniques developed before by Kahn [14], the completion lemma when the dimension of the quasi-linear part is ≥ 1, and the excellence property of an extension given by the function of a 2-fold Pfister form.

2 Preliminaries

From now on, we assume that F is of characteristic 2.

2.1 Notations and definitions

A quadratic form over F of dimension n is a pair (V, φ) where V is an F -vector space of dimension n and φ : V ⟶ F is a map satisfying

(1) φ(αv) = α^{2}φ(v) for any α ∈ F and v ∈ V.

(2) The map B_{φ} : V × V ⟶ F given by B_{φ}(v, w) = φ(v + w) − φ(v) − φ(w) is bilinear (symmetric).

The radical of B_{φ} is the F -vector space rad(B_{φ}) = {v ∈ V | B_{φ}(v, V) = 0}. Since B_{φ} is alternating, i.e. B_{φ}(v, v) = 0 for any v ∈ V, the integer n − dim_{F} rad(B_{φ}) is even. If we set dim_{F} rad(B_{φ}) = s and n − s = 2r, then after a choice of an F -basis of V, the quadratic form φ can be written up to isometry:

φ ≃ [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{r}, b_{r}] ⊥ ⟨c_{1}⟩ ⊥ ⋯ ⊥ ⟨c_{s}⟩ ,

where [a, b] (resp. ⟨c⟩) denotes the quadratic form ax^{2} + xy + by^{2} (resp. the quadratic form cx^{2}). The quadratic form ⟨c_{1}⟩ ⊥ ⋯ ⊥ ⟨c_{s}⟩ is nothing but the restriction of φ to its radical. It is therefore unique up to isometry, and thus the pair (r, s) is also unique. We call this form (resp. the pair (r, s)) the quasilinear part of φ and denote it by ql(φ) (resp. the type of φ).

To shorten notations, we will write ⟨c_{1}, ⋯, c_{s}⟩ instead of ⟨c_{1}⟩ ⊥ ⋯ ⊥ ⟨c_{s}⟩.

Definition 2.1 A quadratic form φ is called:

(1) nonsingular if ql(φ) = 0, and singular otherwise;

(2) totally singular if φ = ql(φ);

(3) nondefective if it is nonsingular or singular with anisotropic quasilinear part.

Let W_{q}(F) (resp. W (F)) denote the Witt group of nonsingular quadratic forms (resp. the Witt ring of regular symmetric bilinear forms).

For a quadratic form φ, we denote by C(φ) its Clifford algebra. If φ is nonsingular, then C(φ) is a central simple algebra over F which, up to isomorphism, only depends on the isometry class of φ, and the center Z(φ) of the even Clifford algebra C_{0}(φ) is a separable quadratic algebra over F, i.e. there exists δ ∈ F with Z(φ) = F [x]/(x^{2} + x + δ). If we put ℘(a) = a^{2} + a, then ℘(F) = {℘(a) | a ∈ F } is an additive subgroup of F, and the class of δ modulo ℘(F) is an invariant of the isometry class of φ called the Arf-invariant of φ and denoted by Δ(φ). More explicitly, if φ ≃ a_{1}[1, b_{1}] ⊥ ⋯ ⊥ a_{n}[1, b_{n}], then Δ(φ) = b_{1} + ⋯ + b_{n} ∈ F/℘(F), and C(φ) = [b_{1}, a_{1}) ⊗_{F} ⋯ ⊗_{F} [b_{n}, a_{n}), where [b, a) (a ∈ F^{∗}, b ∈ F) denotes the quaternion F -algebra generated by two elements u, v over F subject to the relations: u^{2} = a, v^{2} + v = b, uv = (v + 1)u. By abuse of notation, we will often identify Δ(φ) with a representative in F of its class in F/℘(F).

For a quadratic form φ over F and a field extension K/F, we denote by D_{F} (φ) the set of scalars in F^{∗} represented by φ, and by φ_{K} the quadratic form φ ⊗ K. Two quadratic forms φ and ψ are called similar if φ ≃ aψ for some scalar a ∈ F^{∗}.

2.2 Witt decomposition and Witt equivalence

For an integer n ≥ 0 and a quadratic form φ, we denote by n×φ the quadratic form \underbrace{φ ⊥ ⋯ ⊥ φ}_{n times}. A quadratic form φ is called split if φ ≃ r × ℍ ⊥ s ×

⟨0⟩ where ℍ = [0, 0] is the hyperbolic plane. A hyperbolic form is a split nonsingular quadratic form.

A basic notion that we will use in the case of singular forms is the existence of the anisotropic part:

Proposition 2.2 ([10, Prop. 2.4]) If φ is a quadratic form which is not split, then there exists up to isometry a unique anisotropic quadratic form φ_{an} such that

φ ≃ φ_{an} ⊥ r × ℍ ⊥ s × ⟨0⟩ .

This proposition is a consequence of Witt cancellation concerning nonsingular forms and split totally singular forms:

Proposition 2.3 ([17, Prop. 1.2] for (1); [10, Lem. 2.6] for (2)) Let ψ and ψ′ be two quadratic forms (singular or not) such that dim ψ = dim ψ′. Assume that one of the following conditions holds:

(1) φ ⊥ ψ ≃ φ ⊥ ψ′ for some φ ∈ W_{q}(F);

(2) ψ and ψ′ are nondefective and ψ ⊥ s × ⟨0⟩ ≃ ψ′ ⊥ s × ⟨0⟩ for some integer s ≥ 0.

Then, ψ ≃ ψ′.

Definition 2.4 With the same notations as in Proposition 2.2, we call

(1) φ_{an} the anisotropic part of φ;

(2) r × ℍ ⊥ s × ⟨0⟩ the split part of φ;

(3) φ_{an} ⊥ r × ℍ the nondefective part of φ, denoted by φ_{nd};

(4) r (resp. s) the Witt index of φ, denoted by i_{W} (φ) (resp. the defect index or simply defect of φ, denoted by i_{d}(φ));

(5) i_{W} (φ) + i_{d}(φ) the total index of φ, denoted by i_{t}(φ).

Furthermore, we call two forms φ and ψ Witt-equivalent, φ ∼ ψ, if φ_{an} ≃ ψ_{an} (cf. [10]).

Remark 2.5 (i) One readily verifies that i_{t}(φ) is in fact the dimension of a maximal totally isotropic subspace of the underlying vector space V of φ, where a subspace W ⊂ V is said to be totally isotropic if φ(x) = 0 for all x ∈ W.

(ii) If φ and ψ are forms with i_{d}(φ) = i_{d}(ψ) (so in particular if both forms are nondefective), then φ ∼ ψ iff φ ⊥ m × ℍ ≃ ψ ⊥ n × ℍ for some nonnegative integers m, n. □

2.3 The function field of a quadratic form

Let φ be a nonzero quadratic form over F with underlying vector space V of dimension n ≥ 1, and let P_{φ} ∈ F [X_{1}, ⋯, X_{n}] be the homogeneous polynomial of degree 2 corresponding to φ after a choice of a basis of V. The polynomial P_{φ} is reducible if and only if φ_{nd} is either of type (0, 1) or of type (1, 0) and φ_{nd} ≃ ℍ [22, Prop. 3]. When P_{φ} is irreducible, we define the function field F (φ) of φ to be the quotient field of F [X_{1}, ⋯, X_{n}]/(P_{φ}). We set F (φ) = F if P_{φ} is reducible or φ = 0. The extension F (φ)/F is purely transcendental if and only if φ_{nd} is isotropic (i.e. i_{W} (φ) ≥ 1). The polynomial P_{φ} is absolutely irreducible if and only if φ is of type (r, s) with r ≥ 1 and dim φ_{nd} ≥ 3 [1]. If φ is of this shape, then we will say that φ is nice. Therefore, a nice form φ over F has the property that dim(φ_{K})_{nd} ≥ 3 for any field extension K/F, i.e. the property of being a nice form is invariant under field extensions, and the free compositum K · F (φ) can be identified with K(φ), the function field of φ_{K} over K. We also note that F (φ) can in an obvious way be considered as a purely transcendental extension of transcendence degree i_{d}(φ) over F (φ_{nd}).

The function field of a regular symmetric bilinear form B is defined as the function field of the totally singular form v ↦ B(v, v).

Proposition 2.6 ([18, Cor. 3.3]) Let φ and ψ be anisotropic quadratic forms over F. If φ is totally singular and ψ is not totally singular, then φ_{F (ψ)} is anisotropic.

As a corollary we get:

Corollary 2.7 Let φ, ψ be anisotropic quadratic forms over F.

(1) If both φ and ψ are not totally singular, and if φ_{F (ψ)} is isotropic, then i_{W} (φ_{F (ψ)}) ≥ 1.

(2)

i_{t}(φ_{F (φ)}) = \begin{cases}i_{W} (φ_{F (φ)}) & if φ is not totally singular, \\ i_{d}(φ_{F (φ)}) & otherwise. \end{cases}

Proof. (1) If ql(φ) = 0, then there is nothing to prove. If not, Proposition 2.6 implies that ql(φ)_{F (ψ)} is anisotropic. By the uniqueness of the quasilinear part, we necessarily have i_{W} (φ_{F (ψ)}) ≥ 1.

(2) is a direct consequence of Proposition 2.6 and the uniqueness of the quasilinear part. □

2.4 The domination relation and the subform theorem

The subform relation refers to one form being isometric to an orthogonal summand of another, and it is an important tool in the algebraic theory of quadratic forms in characteristic ≠ 2, but it turns out to be too restrictive in characteristic 2 if one wants to study also forms which are singular. It therefore becomes necessary to introduce what we call the domination relation which basically says that one form is simply the restriction to a subspace of the underlyinhg vector space of another form.

Definition 2.8 Let φ and ψ be quadratic forms over F with underlying vector spaces V and W, respectively.

(1) φ is a subform of ψ, denoted by φ ⊂ ψ, if ψ ≃ φ ⊥ φ′ for some quadratic form φ′.

(2) φ is dominated by ψ, denoted by φ ≺ ψ, if there exists an injective isometry t : (φ, V) ⟶ (ψ, W) (i.e. t is an injective F -linear map V → W with ψ(t(v)) = φ(v) for all v ∈ V).

(3) ψ is a nonsingular completion of φ if ψ is nonsingular, φ ≺ ψ and dim ψ = 2r + 2s where (r, s) is the type of φ.

(4) φ is weakly dominated by ψ if aφ ≺ ψ for some a ∈ F^{∗}.

In the following lemma we give an equivalent description of the domination relation.

Lemma 2.9 ([10, Lem. 3.1]) With the same notation as in Definition 2.8, the following are equivalent:

(1) φ ≺ ψ;

(2) There exist nonsingular forms φ_{r} and ψ′, nonnegative integers s′ ≤ s ≤ s′′, c_{i} ∈ F (1 ≤ i ≤ s′′) and d_{j} ∈ F (1 ≤ j ≤ s′) such that φ ≃ φ_{r} ⊥ ⟨c_{1}, ⋯, c_{s}⟩ and

ψ ≃ φ_{r} ⊥ ψ′ ⊥ [c_{1}, d_{1}] ⊥ ⋯ ⊥ [c_{s′}, d_{s′}] ⊥ ⟨c_{s′+1}, ⋯, c_{s′′}⟩ .

Remark 2.10 (1) Clearly, if φ ≺ ψ, dim φ ≥ 2, then ψ_{F (φ)} is isotropic.

(2) Lemma 2.9 implies that if φ ≺ ψ and φ ∈ W_{q}(F), then φ ⊂ ψ.

(3) As was mentioned in [10, Rem. 3.2], the form φ_{r} in Lemma 2.9 can be any nonsingular form with the property that φ ≃ φ_{r} ⊥ ql(φ). □

The following lemma is well-known in characteristic different from 2 for subforms (cf. [9, Lem. 3]).

Lemma 2.11 Let φ and ψ be quadratic forms such that i_{t}(φ) ≥ 1 and ψ ≺ φ. If dim ψ ≥ dim φ − i_{t}(φ) + 1, then ψ is isotropic.

Proof. Let V be the underlying vector space of φ. We may assume that ψ is given by the restriction of φ to some subspace U of V with dim U = dim ψ. By assumption, V contains a totally isotropic subspace W of dimension i_{t}(φ), and also dim U + dim W > dim V. Hence U intersects W non trivially, and thus ψ is isotropic. □

The next lemma will be used frequently in our proofs:

Lemma 2.12 Let S be a totally singular form of dimension s.

(1) If S′ is any nonsingular completion of S, then S ⊥ S′ ≃ S ⊥ s × ℍ.

(2) s × ℍ is a nonsingular completion of S.

Proof. We use the facts that ⟨c⟩ ⊥ [c, d] ≃ ⟨c⟩ ⊥ ℍ and ⟨c⟩ ≺ [c, 0] ≃ ℍ for any c, d ∈ F. □

The analogue of the Cassels-Pfister subform theorem is as follows:

Proposition 2.13 ([18], [10, Th. 4.2]) Let φ, ψ be quadratic forms over F such that φ is nonsingular anisotropic, and ψ is nondefective (singular or not). If φ_{F (ψ)} is hyperbolic, then αβψ ≺ φ for each α ∈ D_{F} (ψ) and β ∈ D_{F} (φ). In particular, dim φ ≥ dim ψ.

The following result is very useful in the proof of Theorem 1.1 since it allows us to avoid the use of Witt cancellation which generally does not hold for singular forms.

Completion Lemma 2.14 ([10, Cor.3.10]) Let φ, ψ be nonsingular quadratic forms over F such that dim φ = dim ψ and φ ⊥ σ ⊥ τ ≃ ψ ⊥ σ ⊥ τ for some totally singular forms σ and τ. Let ρ be a nonsingular completion of σ. Then, there exists a nonsingular completion ρ′ of σ such that φ ⊥ ρ ⊥ τ ≃ ψ ⊥ ρ′ ⊥ τ.

Corollary 2.15 Let φ, ψ be nonsingular quadratic forms over F, and let σ be a totally singular form such that φ ⊥ σ ≃ (dim σ) × ℍ ⊥ ψ ⊥ σ. Then, ψ ⊥ σ ≺ φ.

Proof. The form (dim σ) × ℍ is a nonsingular completion of σ. By the completion lemma there exists a nonsingular completion ρ of σ such that φ ⊥ (dim σ)×ℍ ≃ (dim σ)×ℍ ⊥ ψ ⊥ ρ. Hence, by Witt cancellation (Proposition 2.3(1)) we get φ ≃ ψ ⊥ ρ, and thus ψ ⊥ σ ≺ φ. □

The following corollary can be also deduced from a more general statement in [10, Prop. 3.11]. For the reader’s convenience we give a proof independent of [10].

Corollary 2.16 Let φ and ψ be quadratic forms with φ nonsingular. Then, ψ ≺ φ if and only if i_{W} (φ ⊥ ψ) ≥ dim ψ. If furthermore φ is anisotropic, then ψ ≺ φ if and only if i_{W} (φ ⊥ ψ) = dim ψ.

Proof. Let R be nonsingular such that ψ ≃ R ⊥ ql(ψ). Suppose that ψ ≺ φ and let ψ′ be nonsingular and S be a nonsingular completion of ql(ψ) such that φ ≃ R ⊥ S ⊥ ψ′. Since R ⊥ R ≃ (dim R) × ℍ and ql(ψ) ⊥ S ∼ ql(ψ) we get

φ ⊥ ψ ≃ (dim ψ) × ℍ ⊥ ql(ψ) ⊥ ψ′

and thus i_{W} (φ ⊥ ψ) ≥ dim ψ. If furthermore φ is anisotropic, then necessarily i_{W} (φ ⊥ ψ) ≤ dim ψ by Lemma 2.11, and we have in fact equality.

Conversely, suppose that

φ ⊥ ψ ≃ (dim ψ) × ℍ ⊥ ql(ψ) ⊥ ψ′

for some quadratic form ψ′ ∈ W_{q}(F), then

φ ⊥ R ⊥ ψ ≃ (dim ψ) × ℍ ⊥ R ⊥ ql(ψ) ⊥ ψ′.

By Witt cancellation (Proposition 2.3(1)), we get

φ ⊥ ql(ψ) ≃ (dim ql(ψ)) × ℍ ⊥ ψ ⊥ ψ′.

It follows from Corollary 2.15 that ψ ≺ φ. □

2.5 Pfister neighbors

It is well-known that W_{q}(F) is endowed with a W (F)-module structure as follows. If (V, φ) is a nonsingular quadratic form and B is a regular symmetric bilinear form defined over an F -vector space W, we define a nonsingular quadratic form B ⊗ φ on W ⊗_{F} V by B ⊗ φ(w ⊗ v) = B(w, w)φ(v) whose associated symmetric bilinear form is B ⊗ B_{φ} [7].

Let us write ⟨a_{1}, ⋯, a_{n}⟩_{b} for the bilinear form ∑_{i=1}^{n} a_{i}x_{i}y_{i}, where a_{1}, ⋯, a_{n} ∈ F^{∗}. An n-fold bilinear Pfister form is a form of type ⟨1, a_{1}⟩_{b} ⊗ ⋯ ⊗ ⟨1, a_{n}⟩_{b}. These forms generate the ideal I^{n}F ⊂ W F, where I^{n}F = (IF)^{n} is the n-th power of the ideal of even-dimensional symmetric bilinear forms in W F.

An (n+1)-fold (quadratic) Pfister form is a nonsingular quadratic form of type ⟨1, a_{1}⟩_{b}⊗⋯⊗⟨1, a_{n}⟩_{b}⊗[1, b] for some a_{i} ∈ F^{∗}, b ∈ F. We write ⟨⟨a_{1}, ⋯, a_{n}, b]] for short. ⟨⟨b]] = [1, b] is thus a 1-fold Pfister form. The set of forms isometric (resp. similar) to n-fold Pfister forms will be denoted by P_{n}F (resp. GP_{n}F). We denote the W F -submodule of W_{q}F generated by n-fold Pfister forms by I^{n}W_{q}F, so that I^{n}W_{q}F = (I^{n−1}F)W_{q}F, and we obtain a filtration W_{q}F = IW_{q}F ⊃ I^{2}W_{q}F ⊃ · · ·.

Recall that the Hauptsatz of Arason-Pfister asserts that if φ ∈ I^{n}W_{q}F is anisotropic, then dim φ ≥ 2^{n}, and if dim φ = 2^{n}, then φ ∈ GP_{n}F [6].

A quadratic form φ over F is called a Pfister neighbor if there exist π ∈ P_{n}F (with n satisfying 2^{n−1} < dim φ ≤ 2^{n}) and some a ∈ F^{∗} such that aφ ≺ π, in which case we say that φ is a Pfister neighbor of π. Below, we collect some standard results on Pfister neighbors which we will need in the sequel (see [18, Prop. 3.1]).

Proposition 2.17 (1) A totally singular quadratic form cannot be a Pfister

neighbor.

(2) If φ is a Pfister neighbor of π, then (a) π is unique up to isometry;

(b) For any field extension K/F, φ_{K} is isotropic if and only if π_{K} is also isotropic. In particular, if dim φ ≥ 2, then φ_{F (π)} and π_{F (φ)} are isotropic.

(3) If π ∈ P_{n}F is anisotropic, then φ is a Pfister neighbor of π if and only if dim φ > 2^{n−1} and π_{F (φ)} is isotropic.

Definition 2.18 Let K/F be a field extension.

(1) A quadratic form φ over K is defined over F if there exists a quadratic form ψ over F such that φ ≃ ψ_{K}. In this case, we say that φ is defined by ψ (in general the form ψ is not unique).

(2) We say that K/F is excellent if for any quadratic form φ over F the quadratic form (φ_{K})_{an} is defined over F.

In Section 5 we will see that an extension given by the function field of a quadratic form of dimension 2 or of type (1, 1) is excellent. In characteristic ≠ 2, the corresponding result on 2-dimensional forms is well-known, and for 3-dimensional forms it is due to Arason [8] (see also Rost [25]).

In certain cases, Pfister neighbors can be characterized by their behaviour over their own function field. Also, Pfister forms can be characterized (up to similarity) as being those forms which become hyperbolic over the function field of a form which they dominate provided the dimension of this dominated form is large enough. These are the contents of the following results which will be crucial in the proof of Theorem 1.2:

Theorem 2.19 Let φ be an anisotropic quadratic form of type (r, s) over F.

(i) ([10, Th. 6.6]) Suppose that (φ_{F (φ)})_{an} is defined over F and that 2r > s. If s ≤ 4, or if s = 5 and ql(φ) is similar to ⟨1, a, b, ab⟩ ⊥ ⟨c⟩ for some a, b, c ∈ F^{∗}, then φ is a Pfister neighbor.

(ii) ([10, Cor. 5.3]) Let q be a nonsingular and anisotropic form over F. Suppose that 3 dim φ + s > dim q. If q_{F (φ)} is hyperbolic, then q is similar to a Pfister form. In particular, if 2 dim φ > dim q, then φ is a Pfister neighbor.

2.6 The standard splitting of a quadratic form

For a quadratic form φ with dim φ_{an} ≥ 2, we define its standard splitting tower (F_{i}, φ_{i})_{0≤i≤h} as follows: F_{0} = F, φ_{0} = φ_{an} and for n ≥ 1, we define by induction F_{n} = F_{n−1}(φ_{n−1}) and φ_{n} = ((φ_{n−1})_{F_{n}})_{an}. The smallest integer h for which dim φ_{h} ≤ 1 is called the standard height of φ and denoted by h(φ).

Suppose now that φ is nonsingular. Then the degree deg(φ) of φ is defined as follows. We have h(φ) ≥ 1 and φ_{h(φ)−1} becomes hyperbolic over its own function field, which implies that φ_{h(φ)−1} is similar to an n-fold Pfister form π over F_{h(φ)−1} for some n ≥ 1. We then put deg(φ) = n and we call π the leading form of φ. In the case of a split nonsingular form, we set deg(φ) = ∞.

The set J_{n}F = {φ ∈ W_{q}F | deg(φ) ≥ n} is a W F -submodule of W_{q}F with I^{n}W_{q}F ⊂ J_{n}F. Aravire and Baeza have shown in [4], [5] that in fact J_{n}F = I^{n}W_{q}F. It should be noted that the corresponding result I^{n}F = J_{n}F in characteristic different from 2 is known to be true for n ≤ 5 due to the work on the Milnor Conjecture in small degrees by Merkurjev-Suslin and Rost (see, e.g. [13, Th. 2.8, Remarque]), and for all n due to Orlov-Vishik-Voevodsky [24] (see also [3, Th. 1.5]).

For some results concerning quadratic forms with maximal splitting we will need a generic property of the standard splitting tower of a form which is not totally singular.

Proposition 2.20 Let φ be a form over F which is anisotropic and not totally singular, and let (F_{i}, φ_{i})_{0≤i≤h(φ)} be its standard splitting tower. Let K/F be a field extension such that i_{W} (φ_{K}) ≥ 1. Then, there exists j_{0} ∈ [1, h(φ)] such that

(1) i_{W} (φ_{F_{j_{0}}}) = i_{W} (φ_{K});

(2) Any anisotropic quadratic form over F which is anisotropic over K stays anisotropic over F_{j_{0}}.

Proof. Set k = i_{W} (φ_{K}). Without loss of generality we may suppose dim φ ≥ 3, so that φ is a nice form (cf. subsection 2.3). Since k ≥ 1, the extension K ·F_{1}/K is purely transcendental. Let j_{0} ∈ [1, h(φ)] be maximal such that the extension K · F_{j_{0}}/K is purely transcendental. Since i_{W} (φ_{K·F_{j_{0}}}) = i_{W} (φ_{K}), we get i_{W} (φ_{F_{j_{0}}}) ≤ i_{W} (φ_{K·F_{j_{0}}}) = i_{W} (φ_{K}) = k. If k > i_{W} (φ_{F_{j_{0}}}), then the extension K · F_{j_{0}+1}/K · F_{j_{0}} is purely transcendental, and thus K · F_{j_{0}+1}/K is purely transcendental too, a contradiction to the choice of j_{0}. Hence, k = i_{W} (φ_{F_{j_{0}}}). Now, if ψ is an anisotropic quadratic form over F which is anisotropic over K, then it stays anisotropic over K · F_{j_{0}}, and thus it is also anisotropic over F_{j_{0}} since F_{j_{0}} ⊂ K · F_{j_{0}}. □

Corollary 2.21 With the same notations and hypotheses as in Proposition 2.20, we have i_{W} (φ_{K}) ≥ i_{W} (φ_{F (φ)}).

3 Domination of quadratic forms by Pfister forms and proofs of Theorems 1.1 and 1.3

3.1 Proof of Theorem 1.1

This proof is mainly based on the following proposition which generalizes [20, Prop. 3] to characteristic 2:

Proposition 3.1 Let n ≥ 1 be an integer and let φ be an anisotropic quadratic form of type (r, s) such that

(A) dim φ ≤ 2^{n}, or (B) dim φ = 2^{n} + 1 and r ≥ 1 (i.e. φ is not totally singular).

Then there exists a field extension K/F with an π ∈ P_{n+1}Kanisotropic such that:

(i) φ_{K} ≺ π; (ii) Any anisotropic quadratic form over F stays anisotropic over K(π) if we

are in case (A).

Proof. Let R be a nonsingular form over F such that φ = R ⊥ ql(φ). By assumption, dim R = 2r, dim ql(φ) = s. Let L = F (x_{1}, ⋯, x_{n+1}) be the rational function field in the variables x_{1}, ⋯, x_{n+1} over F, and π = ⟨⟨x_{1}, ⋯, x_{n}, x_{n+1}]]. Now π is a nice form (cf. subsection 2.3), and since 1 ≤ i_{W} (π_{L(\sqrt{x_{1}})}), we have that L(\sqrt{x_{1}})(π)/L(\sqrt{x_{1}}) is purely transcendental. Clearly the extension L(\sqrt{x_{1}})/F is also purely transcendental. Hence, any anisotropic quadratic form over F stays anisotropic over L(π).

Let E/L be a field extension satisfying the conditions:

(C1) π_{E} is anisotropic; (C2) Any anisotropic quadratic form over F stays anisotropic over E(π) if we

are in case (A) of the proposition; (C3) φ_{E} is anisotropic if we are in case (B) of the proposition.

Obviously the field L satisfies (C1) and we have seen that it also satisfies (C2) resp. (C3).

From the uniqueness of the quasilinear part, we get ql(π_{E} ⊥ φ_{E}) ≃ ql(φ)_{E} for any field extension E/F. Hence, if E satisfies (C2) resp. (C3), then ql(φ)_{E} is anisotropic (and in fact also ql(φ)_{E(π)}), and thus i_{t}(π_{E} ⊥ φ_{E}) = i_{W} (π_{E} ⊥ φ_{E}). So for such a field extension, let m(E) = i_{W} (π_{E} ⊥ φ_{E}) and

m = max{ m(E) | E/L satisfies (C1) and [(C2) resp. (C3)]}.

Let K/L be a field extension for which m(K) = m. One has m ≤ dim φ, otherwise π_{K} would be isotropic by Lemma 2.11. If we prove m = dim φ, then Corollary 2.16 implies φ_{K} ≺ π_{K}, and the proof is complete.

Assume now that m < dim φ and set

π_{K} ⊥ R_{K} ⊥ ql(φ)_{K} ≃ ν ⊥ ql(φ)_{K} ⊥ m × ℍ  (1)

for some nonsingular form ν over K and with α := ν ⊥ ql(φ)_{K} anisotropic. We have 2^{n} + 1 ≥ dim φ > m, which, together with (1), implies :

dim α = 2^{n+1} + dim φ − 2m > 2^{n+1} − m > 2^{n} − 1 ≥ 1 ,

hence, dim α ≥ 3. Furthermore, dim ν > 0 because otherwise, again by the relation (1), we would get 2m = 2^{n+1} + 2r, hence m ≥ 2^{n} ≥ dim φ in case (A), and m ≥ 2^{n} + 1 ≥ dim φ in case (B), a contradiction in both cases. This shows that α is a nice form.

By Corollary 2.7, one gets i_{W} (α_{K(α)}) ≥ 1 and thus m(K(α)) = m(K) + i_{W} (α_{K(α)}) ≥ m + 1 > m. To get a contradiction it suffices to prove that the field K(α) satisfies conditions (C1) and [(C2) resp. (C3)].

Property (C1). Adding π_{K} on both sides in relation (1), we get that π_{K} ⊥ α ∼ φ_{K}, and comparing dimensions shows that π_{K} ⊥ α is isotropic. Then, by the anisotropy of π_{K} and α, there exists r ∈ D_{K}(π_{K}) ∩ D_{K}(α). Now suppose that π_{K(α)} is isotropic and hence hyperbolic. By the subform theorem (Proposition 2.13) and Remarks 2.10(3), we obtain r^{2}α ≃ α ≺ π_{K}, and thus

π_{K} ≃ ν ⊥ S ⊥ μ  (2)

for some nonsingular form μ over K and a nonsingular completion S of ql(φ)_{K}. We substitute the relation (2) in the relation (1), we use the fact that S ⊥ ql(φ)_{K} ≃ s × ℍ ⊥ ql(φ)_{K} (Lemma 2.12(1)), and we use Witt cancellation to get

ql(φ)_{K} ⊥ m × ℍ ≃ s × ℍ ⊥ μ ⊥ φ_{K} .

Since s × ℍ is a nonsingular completion of ql(φ)_{K} (Lemma 2.12(2)), the Completion Lemma yields a nonsingular completion S′ of ql(φ)_{K} such that

(m + s) × ℍ ≃ s × ℍ ⊥ μ ⊥ R_{K} ⊥ S′ .

In particular, we have dim(μ ⊥ R_{K} ⊥ S′) = 2m and i_{W} (μ ⊥ R_{K} ⊥ S′) = i_{t}(μ ⊥ R_{K} ⊥ S′) = m. But φ_{K} ≺ μ ⊥ R_{K} ⊥ S′ and dim φ_{K} > m = dim(μ ⊥ R_{K} ⊥ S′) − i_{t}(μ ⊥ R_{K} ⊥ S′), hence φ_{K} is isotropic by Lemma 2.11, a contradiction. Consequently, we have that π_{K(α)} is anisotropic.

Property (C2) in case (A). Let ψ be an anisotropic quadratic form over F. We extend (1) to the field K(π). Since π_{K(π)} is hyperbolic, i.e. i_{W} (π_{K(π)}) = 2^{n} > m, we can use Witt cancellation to get that i_{W} (α_{K(π)}) ≥ 1. Now α is also nice, hence K(π)(α)/K(π) is purely transcendental. If ψ_{K(α)} is isotropic, then also ψ_{K(α)(π)} and hence ψ_{K(π)} is isotropic too, a contradiction to property (C2) for K.

Property (C3) in case (B). We have dim φ = 2^{n} +1, so that φ is singular, and we can write φ ≃ η ⊥ ⟨a⟩ for some form η over F and some a ∈ F^{∗}. Using the proof from case (A), there exists an extension M/L such that π_{M} is anisotropic and η_{M} ≺ π_{M}, so that i_{W} (π_{M} ⊥ φ_{M}) ≥ i_{W} (π_{M} ⊥ η_{M}) = 2^{n} (Corollary 2.16). On the other hand, the maximality of m and the assumption that m < dim φ = 2^{n} + 1 therefore imply that m = 2^{n}. Hence, we obtain

π_{K(α)} ⊥ φ_{K(α)} ≃ α_{K(α)} ⊥ 2^{n} × ℍ ,

and since i_{W} (α_{K(α)}) ≥ 1, we have i_{W} (π_{K(α)} ⊥ φ_{K(α)}) ≥ 2^{n} + 1 = dim φ. By Corollary 2.16, φ_{K(α)} ≺ π_{K(α)}, implying in particular that φ_{K(α)} is anisotropic because K(α) has property (C1). □

Proof of Theorem 1.1. Let φ and ψ be as in Theorem 1.1. Assume that there exists an integer n ≥ 1 such that dim φ ≤ 2^{n} < dim ψ. By Proposition 3.1, there exists a field extension K/F and an anisotropic quadratic form π ∈ P_{n+1}K such that: φ_{K} ≺ π and any anisotropic quadratic form over F stays anisotropic over K(π). If φ_{F (ψ)} is isotropic, then π_{K(ψ)} is also isotropic. Since dim ψ > 2^{n}, it follows from Proposition 2.17(3) that ψ_{K} is a Pfister neighbor of π, and Proposition 2.17(2) implies that ψ_{K(π)} is isotropic, a contradiction. Hence, φ_{F (ψ)} is anisotropic. □

3.2 Proof of Theorem 1.3

To prove this theorem, we need the following result which is a more precise version of case (B) in Proposition 3.1.

Proposition 3.2 Let φ be an anisotropic not totally singular quadratic form of dimension 2^{n}+1 for some integer n ≥ 1. Then, there exists a field extension E/F and an anisotropic π ∈ P_{n+1}E such that

(1) φ_{E} ≺ π, and

(2) the extension E(φ)/F (φ) is unirational (i.e. there exists a purely transcendental extension M/F (φ) such that E(φ) ⊂ M).

Proof. Let x_{1}, ⋯, x_{n+1} be variables over F (φ), L = F (x_{1}, ⋯, x_{n+1}) and π = ⟨⟨x_{1}, ⋯, x_{n+1}]]. Let δ := π ⊥ φ_{L} and (δ_{i}, L_{i})_{0≤i≤h(δ)} its standard splitting tower. By Proposition 3.1, there exists a field extension K/L such that π_{K} is anisotropic and φ_{K} ≺ π_{K}. It follows from Corollary 2.16 that i_{W} (π_{K} ⊥ φ_{K}) = dim φ. By Proposition 2.20 there exists j_{0} ∈ [1, h(δ)] such that

• i_{W} (δ_{L_{j_{0}}}) = 2^{n} + 1, and

• any anisotropic quadratic form over L which is anisotropic over K stays anisotropic over E := L_{j_{0}}.

In particular, π_{E} and φ_{E} are anisotropic. By Corollary 2.16, the condition i_{W} (δ_{E}) = 2^{n} + 1 implies φ_{E} ≺ π_{E}. To finish the proof it suffices to show that the extension E(φ)/F (φ) is unirational. In fact, for N := L(\sqrt{x_{1}})(φ) = F (φ)(\sqrt{x_{1}}, x_{2}, ⋯, x_{n+1}), we have i_{W} (δ_{N}) ≥ 2^{n}+1 since i_{W} (φ_{F (φ)}) ≥ 1 (Corollary 2.7) and π_{N} is isotropic and therefore hyperbolic. Hence

dim(δ_{N})_{an} ≤ 2^{n} − 1  (3)

Moreover, if j ≤ j_{0} − 1, then ql(φ)_{L_{j}} is anisotropic as ql(φ)_{E} is anisotropic, and we clearly have i_{W} (δ_{L_{j}}) < i_{W} (δ_{E}) = 2^{n} + 1. Hence, for j ≤ j_{0} − 1, we get

dim(δ_{L_{j}})_{an} = dim δ_{j} ≥ 2^{n} + 1  (4)

Let M_{j} = N · L_{j} for j ≤ j_{0}. It follows from (3) and (4) that i_{W} ((δ_{j})_{M_{j}}) ≥ 1 for every j ≤ j_{0} − 1, and thus the tower M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{j_{0}} is a succession of purely transcendental extensions. Now M_{0} = N · L_{0} = N · L = F (φ)(\sqrt{x_{1}}, x_{2}, ⋯, x_{n+1}), which is purely transcendental over F (φ). Thus, M_{j_{0}} is purely transcendental over F (φ), and since E(φ) ⊂ M_{j_{0}} we get the claim by taking M = M_{j_{0}}. □

Proof of the Theorem 1.3. Let E/F and π be as in Proposition 3.2 and suppose that φ_{F (ψ)} is isotropic. Then, φ_{E(ψ)} is isotropic, and thus π_{E(ψ)} is hyperbolic. By Proposition 2.17 ψ_{E} is a Pfister neighbor of π. In particular, ψ is not totally singular. Since φ_{E} ≺ π, the form ψ_{E(φ)} is isotropic. Then, ψ_{F (φ)} is isotropic since the extension E(φ)/F (φ) is unirational. □

4 Quadratic forms with maximal splitting

Lemma 4.1 Let φ be an anisotropic quadratic form of dimension 2^{n} +m with 0 < m ≤ 2^{n}. Then, i_{t}(φ_{F (φ)}) ≤ m.

Proof. Let ψ ≺ φ with dim ψ = dim φ − i_{t}(φ_{F (φ)}) + 1. By Lemma 2.11 ψ_{F (φ)} is isotropic, and thus Theorem 1.1 implies that dim φ − i_{t}(φ_{F (φ)}) + 1 > 2^{n}, i.e. i_{t}(φ_{F (φ)}) ≤ m. □

Definition 4.2 Let φ be an anisotropic quadratic form of dimension 2^{n} + m with 0 < m ≤ 2^{n}. φ is said to have maximal splitting if i_{t}(φ_{F (φ)}) = m.

In view of Corollary 2.7, the following lemma is nothing but a reformulation of the definition of maximal splitting:

Lemma 4.3 Let φ be an anisotropic quadratic form of dimension 2^{n} +m with 0 < m ≤ 2^{n} and type (r, s). Let (r′, s′) be the type of φ_{1} = (φ_{F (φ)})_{an}. Then, the following are equivalent:

(1) φ has maximal splitting;

dim(φ_{F (φ)})_{an} = \begin{cases}2^{n} − m & if φ is not totally singular, \\ 2^{n} & otherwise; \end{cases}

(3) (r′, s′) = (r − m, s) if φ is not totally singular (i.e. r ≥ 1), and s′ = 2^{n} otherwise.

Here are some examples of quadratic forms with maximal splitting:

Proposition 4.4 Let φ = R ⊥ ql(φ) be anisotropic.

(1) If dim φ = 2^{n} + 1 form some n ≥ 0, then φ has maximal splitting.

(2) If φ is a Pfister neighbor, and if S is a nonsingular completion of ql(φ) and φ′ is a nonsingular form over F such that R ⊥ S ⊥ φ′ is similar to a Pfister form, then:

(i) (φ_{F (φ)})_{an} ≃ (ql(φ) ⊥ φ′)_{F (φ)}, and (ii) φ has maximal splitting.

Proof. (1) This is a direct consequence of Lemma 4.1.

(2) (i) [10, Prop. 6.1].

(2) (ii) Set dim φ = 2^{n} + m with 0 < m ≤ 2^{n}. Since φ is not totally singular i_{t}(φ_{F (φ)}) = i_{W} (φ_{F (φ)}). Statement (2)(i) implies that 2i_{W} (φ_{F (φ)}) = dim R − dim φ′. Since φ is a Pfister neighbor of an (n + 1)-fold Pfister form, dim φ′ = 2^{n+1} − dim R − 2 dim ql(φ). Hence, i_{W} (φ_{F (φ)}) = m. □

Let φ be an anisotropic form over F. Statement (2)(i) above shows that if φ a Pfister neighbor, then (φ_{F (φ)})_{an} is defined over F. In characteristic not 2, the converse is known to be true due to a result by Knebusch, i.e. if (φ_{F (φ)})_{an} is defined over F, then φ is a Pfister neighbor. In characteristic 2, however, the converse fails to be true in general. A discussion of this problem with various counterexamples can be found in [10].

Now we give two principal results concerning the behaviour of maximal splitting over the function field of a suitable quadratic form:

Lemma 4.5 Let φ be an anisotropic not totally singular quadratic form of dimension 2^{n} + m with 0 < m ≤ 2^{n}. Let ψ be an anisotropic not totally singular quadratic form of dimension > 2^{n}. Then:

(1) Suppose φ is of type (r, s) with 2r ≥ m. Then φ does not have maximal splitting if and only if there exists a field extension K/F such that i_{W} (φ_{K}) < m.

(2) If K/F is a purely transcendental extension, then φ has maximal splitting if and only if φ_{K} has maximal splitting.

(3) If φ_{F (ψ)} is anisotropic, then φ has maximal splitting if and only if φ_{F (ψ)} has maximal splitting.

Proof. (2) is rather obvious and the proofs of (1) and (3) can essentially be copied from those given for the analogous statements in characteristic ≠ 2, see [9, Lem. 4 and 5] (where (1) is proved first and then used to prove (3)), using the fact that if K/F is a field extension such that i_{W} (φ_{K}) ≥ 1, then i_{W} (φ_{K}) ≥ i_{W} (φ_{F (φ)}) (Corollary 2.21). □

Proposition 4.6 Let φ, ψ be two anisotropic not totally singular quadratic forms of respective dimensions 2^{n} + m and 2^{n} + l with 0 < m, l ≤ 2^{n}. Suppose that φ has maximal splitting and φ_{F (ψ)} is isotropic. Then ψ also has maximal splitting and ψ_{F (φ)} is isotropic as well.

Proof. Let φ′ ≺ φ be of dimension 2^{n} + 1. Since i_{W} (φ_{F (φ)}) = m and φ_{F (ψ)} is isotropic, it follows from Corollary 2.21 that i_{W} (φ_{F (ψ)}) ≥ m and thus φ′_{F (ψ)} is isotropic. Hence, φ′ is not totally singular.

By the proof of Proposition 3.2 there exists a field extension E/F which is obtained as a purely transcendental extension followed by a succession of function fields of quadratic forms of dimension > 2^{n}, such that φ′_{E} is a Pfister neighbor of an anisotropic form π ∈ P_{n+1}E. Since φ′_{E(ψ)} is isotropic, it follows that π_{E(ψ)} is hyperbolic, and thus ψ_{E} is a Pfister neighbor of π. By Proposition 4.4 ψ_{E} has maximal splitting, and inductively by Lemma 4.5(2) and (3), ψ has maximal splitting.

Now φ′_{F (φ)} is isotropic by Lemma 2.11, hence i_{W} (φ′_{F (φ)}) ≥ 1 by Corollary 2.7, therefore F (φ)(φ′)/F (φ) is purely transcendental. To complete the proof, we note that by Theorem 1.3, the isotropy of φ′_{F (ψ)} implies that of ψ_{F (φ′)}. But then ψ is obviously isotropic over F (φ)(φ′), thus also over F (φ) because F (φ)(φ′)/F (φ) is purely transcendental. □

5 Proof of Theorem 1.2

5.1 Preliminary results

First of all we need a weak version of a theorem by Aravire and Baeza [5]. For the reader’s convenience we give a proof:

Proposition 5.1 Let φ ∈ I^{n+1}W_{q}(F), and let K be the function field of an anisotropic bilinear Pfister form ψ = ⟨1, a_{1}⟩_{b} ⊗ ⋯ ⊗ ⟨1, a_{n}⟩_{b} for a_{1}, ⋯, a_{n} ∈ F^{∗}. If φ_{K} becomes hyperbolic, then φ ≡ ⟨⟨a_{1}, ⋯, a_{n}, b]] (mod I^{n+2}W_{q}(F)) for some b ∈ F.

Proof. Let ψ_{1} be the totally singular quadratic form defined by ψ_{1}(v) = ψ(v, v). Without loss of generality, we may assume that φ is anisotropic. The condition φ_{K} ∼ 0 implies that x_{1}ψ_{1} ≺ φ for some x_{1} ∈ F^{∗}. Choose δ a not totally singular form of dimension 2^{n} + 1 such that x_{1}ψ_{1} ≺ δ ≺ φ. One readily verifies that δ is a Pfister neighbor of a Pfister form π_{1} = ⟨⟨a_{1}, ⋯, a_{n}, b_{1}]] for some b_{1} ∈ F.

Since δ is anisotropic, one easily checks that i_{t}(δ ⊥ δ) = dim δ, hence i_{t}(φ ⊥ x_{1}π_{1}) = i_{W} (φ ⊥ x_{1}π_{1}) ≥ 2^{n} + 1 = dim δ because δ ≺ φ and δ ≺ x_{1}π_{1}. Now, let φ_{1} = (φ ⊥ x_{1}π_{1})_{an}. We thus get dim φ_{1} = dim φ + dim π − 2i_{W} (φ ⊥ x_{1}π_{1}) < dim φ. Moreover, (φ_{1})_{K} ∼ 0. By induction on the dimension of φ, we obtain φ ∼⊥_{i=1}^{m} x_{i} ⟨⟨a_{1}, ⋯, a_{n}, b_{i}]] for some integer m and some x_{i} ∈ F^{∗}, b_{i} ∈ F. Since ⟨⟨a_{1}, ⋯, a_{n}, b_{i}]] ≡ x_{i} ⟨⟨a_{1}, ⋯, a_{n}, b_{i}]] (mod I^{n+2}W_{q}(F)), we get the claim by taking b = ∑_{i=1}^{m} b_{i}. □

Remark 5.2 (i) This proof shows that the Witt kernel W_{q}(K/F) is generated as F -module by Pfister forms of type ⟨⟨a_{1}, ⋯, a_{n}, b]], or equivalently W_{q}(K/F) = ψ ⊗ W_{q}(F) (ψ and K as in the proposition).

(ii) For the Witt kernel W_{q}(L/F) where L = F (π) for an anisotropic quadratic Pfister form π ∈ P_{n}F, one obtains W_{q}(L/F) = {ρ ⊗ π | ρ ∈ W (F)}. More precisely, if φ is a nonsingular anisotropic quadratic form with φ_{L} hyperbolic, then φ ≃⊥_{i=1}^{m} c_{i}π for some m and some c_{i} ∈ F^{∗}. This can easily be deduced from Proposition 2.13, using the fact that π_{L} is hyperbolic. □

Lemma 5.3 Let A be a finite dimensional central simple F -algebra, and let L be a quadratic extension given by F [x]/(x^{2}+u) (u ∈ F^{∗}) resp. F [x]/(x^{2}+x+u) such that ind A_{L} ≤ 2. Then there exists a quaternion F -algebra Q and a ∈ F resp. b ∈ F^{∗} such that A = [a, u) ⊗_{F} Q resp. A = [u, b) ⊗_{F} Q ∈ Br(F), according as L/F is inseparable or separable.

Proof. The condition ind A_{L} ≤ 2 implies ind A ≤ 4.

If ind A ≤ 2, then A is Brauer-equivalent to a quaternion algebra Q and the lemma is true by taking for a (resp. for b) any element in ℘(F) (resp. in F^{∗2}).

If ind A = 4, then there exists an Albert form φ such that A = C(φ) ∈ Br(F) (an Albert form is a 6-dimensional nonsingular quadratic form of trivial Arfinvariant). By [21], the form φ is anisotropic and φ_{L} is isotropic. It follows from [18, Th. 1.1] that there exists φ′ of dimension 2, a, c ∈ F and b ∈ F^{∗} such that bφ ≃ [1, c] ⊥ u[1, a] ⊥ φ′ or b[1, u] ⊂ φ. To get the desired claim we take the Clifford algebra and we use the fact that C(φ) = C(bφ) ∈ Br(F). □

Lemma 5.4 Let φ be a quadratic form which is not split, and let L be a quadratic extension. Then, (φ_{L})_{an} is defined over F.

Proof. Let V be the underlying vector space of φ. We distinguish between separable and inseparable quadratic extension. Without loss of generality, we may assume that φ is anisotropic. There is nothing to prove if φ_{L} is anisotropic. So suppose that φ_{L} is isotropic.

Suppose first that L/F is separable, and let a ∈ F^{∗}, α ∈ L^{∗} be such that L = F (α) and α^{2}+α+a = 0. In this case φ is not totally singular as anisotropic totally singular forms stay anisotropic over separable extensions (see, e.g. [10, Prop. 8.7]). The isotropy of φ_{L} implies the existence of v, w ∈ V, not both zero, such that φ(v + αw) = 0, hence φ(v) = aφ(w) and B_{φ}(v, w) = φ(w). Since φ is anisotropic, we get φ(v) ≠ 0 and φ(w) ≠ 0. It follows that φ(w)[1, a] ⊂ φ. If we set φ ≃ φ(w)[1, a] ⊥ φ′, it follows that φ_{L} ∼ φ′_{L}. Since dim φ′ < dim φ the claim follows by induction on dim φ.

Suppose finally that L/F is inseparable, and let b ∈ F^{∗} be such that L = F (\sqrt{b}). In this case the isotropy of φ_{L} implies the existence of v′, w′ ∈ V, not both zero, such that φ(v′) = bφ(w′) and B_{φ}(v′, w′) = 0 (also φ(v′) ≠ 0 and φ(w′) ≠ 0 since φ is anisotropic). It follows that φ(w′) ⟨1, b⟩ ≺ φ. By Lemma 2.9 there exists x, y, z, t ∈ F^{∗} such that φ(w′) ⟨1, b⟩ ≃ ⟨x, y⟩ and one of the three following forms is a subform of φ: ⟨x, y⟩, [x, z] ⊥ ⟨y⟩, or [x, z] ⊥ [y, t]. Since xy ∈ L^{∗2}, we have ⟨x, y⟩_{L} ≃ ⟨x, 0⟩_{L}, ([x, z] ⊥ ⟨y⟩)_{L} ≃ ℍ ⊥ ⟨y⟩_{L}, and ([x, z] ⊥ [y, t])_{L} ≃ [x, z + (yt/x)]_{L} ⊥ ℍ. Hence, one can easily see that in the first case φ_{L} ≃ ⟨0⟩ ⊥ φ′_{L}, and in the other two cases φ_{L} ≃ ℍ ⊥ φ′_{L} for some quadratic form φ′ defined over F. In this case we also conclude by induction on dim φ. □

Remark 5.5 The excellence of quadratic extensions in characteristic 2 (separable or inseparable) has previously been shown in [2]. □

For the excellence of an extension given by the function field of a 2-fold Pfister form we need the following proposition which generalizes [25, Proposition] to characteristic 2:

Proposition 5.6 Let K be the quotient field of F [s, t]/(s^{2} + st + at^{2} + b), and let φ be a quadratic form (singular or not). Then there exists an integer p, quadratic forms φ_{i}, ψ_{i} for 0 ≤ i ≤ p, and scalars c_{1}, ⋯, c_{p−1} ∈ F^{∗} such that φ_{0} = φ and (1) φ_{i} ≃ c_{i}[1, a] ⊥ ψ_{i} for 0 ≤ i ≤ p − 1;

(2) φ_{i+1} ≃ bc_{i}[1, a] ⊥ ψ_{i} for 0 ≤ i ≤ p − 1;

(3) ((φ_{p})_{K})_{an} ≃ ((φ_{p})_{an})_{K}.

By the same reasoning as in [25, Corollary, end of page 511], we obtain the following corollary:

Corollary 5.7 An extension given by the function field of a 2-fold Pfister form is excellent.

Proof of Proposition 5.6. With the same notations as in [25], one can essentially reproduce the proof of [25, Proposition] after applying the following changes:

• For R we take the ring F [s, t]/(s^{2} + st + at^{2} + b);

• In [25, Lemma, page 512], we take [1, a] instead of ⟨1, a⟩;

• The claim in [25, middle of page 512] becomes:

B_{φ}(v_{n}, w_{n}) = φ(v_{n}) and φ(w_{n}) = −aφ(v_{n});

• At the end of [25, page 512] we change L by F [z]/(z^{2} + z + a), and clearly in [25, page 513] the conjugate of λ is given by α ↦ α + 1.

We leave the details to the reader. □

The following corollary is well-known in characteristic different from 2, due to Elman and Lam [8]:

Corollary 5.8 Let τ_{1}, τ_{2} ∈ P_{2}F, and let φ be a nonsingular quadratic form which becomes hyperbolic over F (τ_{1})(τ_{2}). Then, there exists symmetric bilinear forms ρ_{1} and ρ_{2} such that φ ∼ ρ_{1} ⊗ τ_{1} ⊥ ρ_{2} ⊗ τ_{2}.

Proof. Essentially the same proof like the one of [8, Cor 2.12] by using the excellence property given by Corollary 5.7. □

Corollary 5.9 With the same notations and hypotheses as in Corollary 5.8, and if ind C(φ) = 4, then there exist scalars α_{1}, α_{2} ∈ F^{∗} such that φ ≡ α_{1}τ_{1} ⊥ α_{2}τ_{2} (mod I^{4}W_{q}(F)).

Proof. Let ρ_{1}, ρ_{2} be as in Corollary 5.8. Now for a, b ∈ F^{∗}, we have ⟨a, b⟩_{b} ⊗ τ_{i} ∈ I^{3}W_{q}F, and thus C(⟨a, b⟩_{b} ⊗ τ_{i}) ∼ 0 ∈ Br(F) (cf. [26]). Thus, by using the Clifford algebra, we deduce that ρ_{1} and ρ_{2} are of odd dimension, otherwise we would get ind C(φ) ≤ 2. For α_{1}, α_{2} ∈ F^{∗} such that ρ_{1} ⊥ ⟨α_{1}⟩_{b} ∈ I^{2}F and ρ_{2} ⊥ ⟨α_{2}⟩_{b} ∈ I^{2}F, we therefore get φ ≡ α_{1}τ_{1} ⊥ α_{2}τ_{2} (mod I^{4}W_{q}(F)). □

5.2 Proof of Theorem 1.2

Let ψ be as in Theorem 1.2, (r, s) its type, ψ_{1} = (ψ_{F (ψ)})_{an} and K = F (ψ). Set ψ = ψ′ ⊥ ql(ψ) with ψ′ ∈ W_{q}(F), and set ql(ψ) = ⟨c_{1}, ⋯, c_{s}⟩, where we may assume after scaling that c_{1} = 1 if s > 0. By the uniqueness of the quasilinear part, Proposition 2.6 and Lemma 4.3, we have ql(ψ_{1}) = ql(ψ)_{K} and dim ψ_{1} ≤ 6 (where s = 0 if dim ψ_{1} = 6), and thus s ≤ 5. Hence, 2r > s and the quasi-linear part of ψ satisfies the assumption made in Theorem 2.19(i). To conclude, it therefore suffices by Theorem 2.19 to prove that either ψ_{1} is defined over F, or that there exists a nonsingular anisotropic form q over F with 2 dim φ > dim q and q_{F (φ)} hyperbolic.

By Lemma 4.3, dim ψ_{1} = 0 if and only if dim ψ = 2^{n+1}, in which case we are done (a 0-dimensional form is clearly defined over F).

If dim ψ_{1} > 0 and ψ_{1} is totally singular, then ψ_{1} is defined over F by ql(ψ), and again, we are done. So we may suppose that ψ_{1} is not totally singular. Thus, the hypotheses on ψ in Theorem 1.2 together with Lemma 4.3 imply that we are in one of the following cases (where ψ has maximal splitting by assumption):

(a) ψ_{1} is of type (1, 0), or, equivalently, ψ is of dimension 2^{n+1} − 2 and of type (2^{n} − 1, 0);

(b) ψ_{1} is of type (1, 1), or, equivalently, ψ is of dimension 2^{n+1} − 3 and of type (2^{n} − 2, 1);

(c) ψ_{1} is of type (2, 0), or, equivalently, ψ is of dimension 2^{n+1} − 4 and of type (2^{n} − 2, 0);

(d) ψ_{1} is of type (1, 2), or, equivalently, ψ is of dimension 2^{n+1} − 4 and of type (2^{n} − 3, 2);

(e) ψ_{1} is of type (2, 1), or, equivalently, ψ is of dimension 2^{n+1} − 5 and of type (2^{n} − 3, 1);

(f) ψ ∈ I^{2}W_{q}F and ψ_{1} is of type (3, 0), or, equivalently, ψ ∈ I^{2}W_{q}F and ψ is of dimension 2^{n+1} − 6 and of type (2^{n} − 3, 0).

Case (a). Set ψ_{1} = α [1, β]. By comparing Arf-invariants in the relation ψ_{K} ∼ ψ_{1}, we may assume β ∈ F. We follow the argument given in [14, page 149]. Take L = F [x]/(x^{2} + x + β) and M = L(ψ). Since C(ψ)_{M} = [β, α)_{M} = 0 ∈ Br(L) and dim ψ > 4 (because n ≥ 2), we get C(ψ)_{L} = 0 ∈ Br(K). Then C(ψ) is Brauer-equivalent to a quaternion algebra [β, γ) for some γ ∈ F^{∗}. Hence, ⟨⟨α, β]] ≃ (⟨⟨γ, β]])_{K}, and by Witt cancellation α [1, β] ≃ (γ [1, β])_{K}.

Case (b). Set ψ_{1} = α [1, β] ⊥ ⟨1⟩ (recall that c_{1} = 1 and ql(ψ_{1}) = ql(ψ)_{K}). We then get

C_{0}(ψ)_{K} = C(ψ′)_{K} = [β, α) ∈ Br(K)

(see, e.g., [22, Lemma 2]). Since dim ψ ≥ 5 (because n ≥ 2), the index reduction theorem [22] implies that C_{0}(ψ) is Brauer-equivalent to a quaternion algebra [d, e) for some d ∈ F and e ∈ F^{∗}. Hence,

[1, β] ⊥ α [1, β] ≃ ([1, d] ⊥ e [1, d])_{K} .

Adding on both sides the form ⟨1⟩, Witt cancellation yields ⟨1⟩ ⊥ α [1, β] ≃ (⟨1⟩ ⊥ e [1, d])_{K}. The claim then follows.

Case (c). We consider two cases depending on whether ψ_{1} ∈ GP_{2}K or not.

Suppose first that ψ_{1} ∈ GP_{2}K. Then ind C(ψ)_{K} ≤ 2. Since dim ψ ≥ 12 (because n ≥ 3), we get by the index reduction theorem the existence of τ ∈ P_{2}F such that C(ψ) is Brauer-equivalent to C(τ). In particular, ψ_{1} is similar to τ_{K} and thus ψ_{K(τ)} ∼ 0 and therefore ψ_{F (τ)(ψ)} ∼ 0. Also, τ is necessarily anisotropic.

If ψ_{F (τ)} is anisotropic (or, equivalently, if i_{W} (ψ_{F (τ)}) = 0), then we have i_{W} (ψ_{F (τ)(ψ)}) = 2^{n} − 2 > 2^{n} − 4, and dim ψ = 2^{n} + (2^{n} − 4), a contradiction to Lemma 4.1. Hence, i_{W} (ψ_{F (τ)}) > 0 and F (τ)(ψ)/F (τ) is purely transcendental. This implies that ψ_{F (τ)} is already hyperbolic. By Remark 5.2(ii), there exists a symmetric bilinear form ρ such that ψ ≃ ρ ⊗ τ. The dimension of ρ is necessarily odd since C(ψ) = C(τ) ≠ 0 ∈ Br(F). For x ∈ F^{∗} such that ρ ⊥ ⟨x⟩_{b} ∈ I^{2}F, we get ψ ⊥ xτ ∈ I^{4}W_{q}(F). We extend this relation to K and we use the Hauptsatz of Arason-Pfister to get ψ_{1} ≃ (xτ)_{K}.

Now suppose that ψ_{1} ∉ GP_{2}K and let k ∈ F^{∗} be such that Δ(ψ) = k (mod ℘(F)). We follow the argument given in [14, page 149]. Let N = F [x]/(x^{2} + x + k). Note that [N : F] = 2 since otherwise ψ_{1} ∈ GP_{2}K.

Since ind C(ψ)_{N(ψ)} ≤ 2, it follows from the index reduction theorem that ind C(ψ)_{N} ≤ 2. By Lemma 5.3, there exists ρ := ⟨⟨d, c]] ∈ P_{2}F and r ∈ F^{∗} such that C(ψ) is Brauer-equivalent to C(ρ) ⊗_{F} [k, r).

Now let ν = r([1, c + k] ⊥ d [1, c]). It is clear that Δ(ν) = Δ(ψ) (i.e. Δ(ν ⊥ ψ) = 0 (mod℘(F))), and one readily checks that C(ψ ⊥ ν) = 0 ∈ Br(F). By a result of Sah [26], this implies μ := ψ ⊥ ν ∈ I^{3}W_{q}(F).

Also, ν_{N} ≃ rρ_{N} and C(ψ)_{N} = C(ρ)_{N} = C(ν_{N}) ∈ Br(N). Hence, C(ψ_{1})_{N(ψ)} is Brauer-equivalent to C(ν)_{N(ψ)}, and we deduce by [21] that ν_{K} is similar to ψ_{1}. Therefore, K(ψ_{1}) = K(ν), and since dim(ψ_{K(ψ_{1})})_{an}, dim(ν_{K(ν)})_{an} ≤ 2, the Hauptsatz of Arason-Pfister implies that μ_{K(ν)} ∼ 0.

Now dim(ψ_{F (ν)})_{an} ≤ 4 or = 2^{n+1} − 4 by Proposition 2.20. Furthermore, we have dim(ψ_{K(ψ_{1})})_{an} = dim(ψ_{F (ψ)(ν))})_{an} = dim(ψ_{F (ν)(ψ))})_{an} ≤ 2 by the above, and thus, we cannot have dim(ψ_{F (ν)})_{an} = 2^{n+1} − 4 (otherwise, we would get a contradiction to Lemma 4.1). Hence, dim(ψ_{F (ν)})_{an} ≤ 4, and since dim(ν_{F (ν)})_{an} ≤ 2, the Hauptsatz again implies that μ_{F (ν)} is hyperbolic.

Now ν_{F (ρ)} is isotropic as ν weakly dominates the Pfister neighbor ⟨1⟩ ⊥ d[1, c] of ρ ∈ P_{2}F. Hence, the extension F (ρ)(ν)/F (ρ) is purely transcendental and we therefore must have μ_{F (ρ)} ∼ 0. But ν ∼ rρ ⊥ r[1, k], hence, (ψ ⊥ r [1, k])_{F (ρ)} ∼ 0. Since C(ψ ⊥ r [1, k]) is Brauer-equivalent to C(ρ), there exists a symmetric bilinear form λ of odd dimension such that ψ ⊥ r [1, k] ∼ λ ⊗ ρ (see Remark 5.2). Now, for y ∈ F^{∗} such that λ ⊥ ⟨y⟩ ∈ I^{2}F, we get ψ ⊥ r [1, k] ⊥ yρ ∈ I^{4}W_{q}(F). We extend this relation to K to get ψ_{1} ⊥ (r [1, k] ⊥ yρ)_{K} ∈ I^{4}W_{q}(K). The Hauptsatz of Arason-Pfister implies that ψ_{1} ∼ (r [1, k] ⊥ yρ)_{K}. By comparing dimensions and by using Theorem 1.1 we conclude that r [1, k] ⊥ yρ is isotropic, and thus ψ_{1} is defined over F.

Case (d). Let α, β ∈ K^{∗} be such that

(ψ_{K})_{an} ≃ α[1, β] ⊥ ⟨1, c_{2}⟩_{K} .

Let ψ′′ = ψ′ ⊥ ⟨1⟩. We then get ψ_{K(\sqrt{c_{2}})} ∼ ψ′′_{K(\sqrt{c_{2}})} ∼ (α[1, β] ⊥ ⟨1⟩)_{K(\sqrt{c_{2}})} and

C_{0}(ψ′′)_{K(\sqrt{c_{2}})} = C(ψ′)_{K(\sqrt{c_{2}})} = [β, α)_{K(\sqrt{c_{2}})} ∈ Br(K(\sqrt{c_{2}}))

(see, e.g., [22, Lemma 2]). Since K(\sqrt{c_{2}}) is a purely transcendental extension of F (\sqrt{c_{2}})(ψ′′) and dim ψ′′ ≥ 11, we get by the index reduction theorem ind C_{0}(ψ′′)_{F (\sqrt{c_{2}})} ≤ 2. By Lemma 5.3, there exist v ≠ 0, a, b ∈ F such that

C_{0}(ψ′′) = C(ψ′) = [a, c_{2}) ⊗ [b, v) ∈ Br(F).

Now let ℓ ∈ F be such that Δ(ψ′) = l (mod℘(F)), and let

ρ = ψ′ ⊥ [1, a + b + ℓ] ⊥ c_{2}[1, a] ⊥ v[1, b].

Clearly, ψ ≺ ρ and dim ρ = 2^{n+1}. Furthermore, Δ(ρ) = 0 (mod℘(F)), and a quick computation yields C(ρ) = 0 ∈ Br(F), so that ρ ∈ I^{3}W_{q}F. Also, ρ ≁ 0 since dim ψ > \frac{1}{2} dim ρ and ψ is anisotropic. Note that ρ is a nonsingular completion of ψ′ ⊥ v[1, b] ⊥ ⟨1, c_{2}⟩. After passing to K and by invoking the completion lemma, there exist x, y ∈ K such that for the K-form

π := α[1, β] ⊥ [1, x] ⊥ c_{2}[1, y] ⊥ v[1, b]

we have ρ_{K} ∼ π, and thus π ∈ I^{3}W_{q}K. Since dim π = 8, the Hauptsatz of Arason-Pfister yields π ∈ P_{3}K and δ := ⟨1, c_{2}, v⟩ ≺ π.

If δ is isotropic, then π is isotropic and hence hyperbolic, and thus ρ_{K} ∼ 0. By Theorem 2.19(ii), ρ_{an} is similar to a Pfister form of which ψ is a neighbor. A simple dimension count then shows that indeed ρ = ρ_{an}, and it is a Pfister form since 1 ∈ D_{F} (ρ), i.e. ρ ∈ P_{n+1}F.

Now suppose that δ is anisotropic and let δ′ := δ ⊥ ⟨vc_{2}⟩, so that δ′ is the totally singular form derived from the bilinear Pfister form ⟨1, c_{2}⟩_{b} ⊗ ⟨1, v⟩_{b}. It follows from [10, Prop. 8.9(iii)] that for any field extension M/F, we have that δ_{M} is isotropic iff δ′_{M} is isotropic. Hence, δ_{K} and thus also δ′_{K} are anisotropic (Proposition 2.6).

Let E = F (δ′) and L = K(δ′). Then δ_{L} is isotropic, and since δ_{L} ≺ π_{L}, it follows that π_{L} is hyperbolic, and thus ρ_{L} = ρ_{E(ψ)} ∼ 0. Note that ql(ψ_{E}) ≃ ⟨1, c_{2}⟩_{E} is anisotropic by Theorem 1.1 as ⟨1, c_{2}⟩ is anisotropic over F, and E = F (δ′) with δ′ anisotropic of dimension 4. In particular, ψ_{E} is nondefective.

If dim(ρ_{E})_{an} > 0, then ψ_{E} is necessarily anisotropic (otherwise L/E would be purely transcendental, a contradiction to ρ_{E} ≁ 0). By comparing dimensions and by Theorem 2.19(ii), (ρ_{E})_{an} is similar to a Pfister form of which ψ_{E} is a neighbor. This Pfister form is then necessarily of dimension 2^{n+1}. But ρ_{E} is isotropic since δ ≺ ρ and δ_{E} is isotropic. Thus, dim(ρ_{E})_{an} ≤ 2^{n+1} − 2, a contradiction. Hence, ρ_{E} ∼ 0, and by Proposition 5.1, there exists c ∈ F such that

ρ ≡ ⟨⟨c_{2}, v, c]] (modI^{4}W_{q}(F)).

Let

γ = ψ′ ⊥ [1, a + b + c + ℓ] ⊥ c_{2}[1, a + c] ⊥ v[1, b + c] ⊥ c_{2}v[1, c].

We have dim γ = 2^{n+1} +2, γ ∼ ρ ⊥ ⟨⟨c_{2}, v, c]] ∈ I^{4}W_{q}F, and ψ ≺ γ. Since ψ is anisotropic and dim ψ > \frac{1}{2} dim γ, the form γ is not hyperbolic. Let γ′ = γ_{an}. We have γ′_{K} ≡ π ⊥ ⟨⟨c_{2}, v, c]]_{K} (mod I^{4}W_{q}(K)) and dim(π ⊥ ⟨⟨c_{2}, v, c]]_{K})_{an} < 16 since δ ≺ π and δ ≺ ⟨⟨c_{2}, v, c]]. Hence, by the Hauptsatz, γ′_{K} ∼ 0. Since 2 dim ψ > dim γ′, we get again by Theorem 2.19(ii) that γ′ is similar to a Pfister form of which ψ is a neighbor.

The cases (e) and (f). Here, ψ_{1} is of dimension 5 and type (r, 1) in case (e) (in which ql(ψ) = ⟨1⟩), or of dimension 6, nonsingular, with trivial Arfinvariant in case (f) (i.e., ψ_{1} is an Albert form). Set ψ_{1} = ψ_{2} ⊥ ql(ψ)_{K}. Let ρ = ψ′ ⊥ [1, Δ(ψ′)] if dim ψ = 2^{n+1} − 5 (resp. ρ = ψ = ψ′ if dim ψ = 2^{n+1} − 6). In particular, Δ(ρ) is trivial and thus ρ ∈ I^{2}W_{q}F. We follow some arguments of the proof by Izhboldin [11]. Now one readily sees that C(ρ) = C(ψ′) ∈ Br(F) (= C_{0}(ψ) in case (e)), and thus C(ρ)_{K} = C(ψ′)_{K} = C(ψ_{2}) ∈ Br(K). Hence, ind C(ρ)_{K} = ind C(ψ_{2}) ≤ 4, and we get by the index reduction theorem that ind C(ρ) ≤ 4.

\underline{The subcase ind C(ρ) = 4.} Let τ_{1}, τ_{2} ∈ P_{2}F be such that C(ρ) = C(τ_{1}) ⊗_{F} C(τ_{2}) ∈ Br(F). Since C(ρ) is split over L := F (τ_{1})(τ_{2}), we get by a result of Sah [26] that ρ_{L} ∈ I^{3}W_{q}(L). Since dim ψ_{1} ≤ 6 we get dim(ρ_{L(ψ)})_{an} ≤ 6 (in fact, if dim ψ = 2^{n+1} − 5 we get by the completion lemma ρ_{K} ∼ ψ_{2} ⊥ [1, m] for some m ∈ K, and if dim ψ = 2^{n+1} − 6 we also have dim(ρ_{K})_{an} ≤ 6).

By the Hauptsatz, ρ_{L(ψ)} is hyperbolic. If ρ_{L} is not hyperbolic then ψ_{L} is anisotropic, and by Theorem 2.19(ii), (ρ_{L})_{an} is similar to a Pfister form, necessarily of dimension 2^{n+1}, a contradiction to dim ρ < 2^{n+1}. Hence, ρ_{L} ∼ 0. By Corollary 5.9, there exist α_{1}, α_{2} such that ρ ⊥ α_{1}τ_{1} ⊥ α_{2}τ_{2} ∈ I^{4}W_{q}(F). Since λ := (α_{1}τ_{1} ⊥ α_{2}τ_{2})_{an} has dimension ≤ 8, we get dim(ρ_{K})_{an}+dim(λ_{K})_{an} ≤ 14 < 16. It follows from the Hauptsatz that ρ_{K} ∼ λ_{K}. Note that λ_{K} is anisotropic since dim λ ≤ 8 < dim ψ (Theorem 1.1), and thus (ρ_{K})_{an} ≃ λ_{K} which implies dim λ ≤ 6. Note that in the case dim ψ = 2^{n+1} − 5, we have ρ ⊥ ⟨1⟩ ∼ ψ, hence

(ρ ⊥ ⟨1⟩)_{K} ∼ ψ_{1} ∼ (λ ⊥ ⟨1⟩)_{K} .

But the form (λ ⊥ ⟨1⟩)_{an} (which is of dimension < 8 < dim ψ) stays also anisotropic over K by Theorem 1.1, and thus

(ψ_{K})_{an} ≃ \begin{cases}((λ ⊥ ⟨1⟩)_{an})_{K} & if dim ψ = 2^{n+1} − 5 , \\ λ_{K} & if dim ψ = 2^{n+1} − 6 . \end{cases}

\underline{The subcase ind C(ρ) ≤ 2.} By the same arguments as before we get ρ_{F (τ)} ∼ 0 where τ ∈ P_{2}F satisfies C(ρ) = C(τ) ∈ Br(F). By Remark 5.2(ii), there exists a symmetric bilinear form μ such that ρ ∼ μ ⊗ τ. Since C(ρ) = C(τ), the dimension of μ is necessarily odd and thus ρ ⊥ zτ ∈ I^{4}W_{q}(F) where z ∈ F^{∗} satisfies μ ⊥ ⟨z⟩_{b} ∈ I^{2}F. Now it is clear that the proof can be completed as in the previous case. In fact, the form λ in the above proof will now be given by zτ_{an}, which shows in particular that this case cannot occur in the case dim ψ = 2^{n} − 6, and that necessarily ind C(ρ) = 2 if dim ψ = 2^{n} − 5. □

5.3 Proof of Corollary 1.4

By Theorem 1.3, ψ_{F (φ)} is also isotropic and ψ is not totally singular. Since φ has maximal splitting and dim ψ > 2^{n}, we get by Proposition 4.6 that ψ has maximal splitting too, and thus by Theorem 1.2, ψ is a Pfister neighbor of an (n + 1)-fold Pfister form π. Since ψ_{F (φ)} is isotropic and dim φ > 2^{n} we deduce that φ is also a Pfister neighbor of π. □

References

[1] H. Ahmad, The algebraic closure in function fields of quadratic forms in characteristic 2, Bull. Austral. Math. Soc. 55 (1997) 293–297.

[2] H. Ahmad, Witt kernels of bi-quadratic extensions in characteristic 2, preprint.

[3] J.Kr. Arason, R. Elman, Powers of the fundamental ideal in the Witt ring, J. Algebra 239 (2001) 150–160.

[4] R. Aravire, R. Baeza, A note on generic splitting of quadratic forms, Comm. Alg. 27 (1999) 3473–3477.

[5] R. Aravire, R. Baeza, The behaviour of quadratic and differential forms under function field extensions in characteristic 2, J. Algebra 259 (2003) 361–414.

[6] R. Baeza, Ein Teilformensatz für quadratische Formen in Charakteristik 2, Math. Z. 135 (1974) 175–184.

[7] R. Baeza, Quadratic forms over semilocal rings, Lect. Notes Math. vol. 655, Springer, Berlin–Heidelberg–New York 1978.

[8] R. Elman, T.Y. Lam, A. Wadsworth, Amenable fields and Pfister extensions, in: Conference on Quadratic forms 1976 (ed. by G. Orzech), Queen’s Papers in Pure and Appl. Math. 64, Queen’s Univ., Kingston, Ontario, 1977, pp. 445–492.

[9] D.W. Hoffmann, Isotropy of quadratic forms over the function field of a quadric, Math. Z. 220 (1995) 461–476.

[10] D.W. Hoffmann, A. Laghribi, On quadratic forms and Pfister neighbors in characteristic 2, to appear in Trans. Amer. Math. Soc.

[11] O.T. Izhboldin, Quadratic forms with maximal splitting, St. Petersburg Math. J. 9 (1997) 219-224.

[12] O.T. Izhboldin, Motivic equivalence of quadratic forms, II, Manuscr. Math. 102 (2000) 41–52.

[13] B. Kahn, Formes quadratiques de hauteur et de degré 2, Indag. Mathem. N.S. 7 (1996) 47–66.

[14] B. Kahn, A descent problem for quadratic forms, Duke Math. J. 80 (1995), 139–159.

[15] N. Karpenko, On the first Witt index of quadratic forms, Invent. Math. 153 (2003) 455–462.

[16] N. Karpenko, A. Merkurjev, Essential dimension of quadrics, Invent. Math. 153 (2003) 361–372.

[17] M. Knebusch, Specialization of quadratic and symmetric bilinear forms, and a norm theorem, Acta Arithmetica 24 (1973) 279–299.

[18] A. Laghribi, Certaines formes quadratiques de dimension au plus 6 et corps des fonctions en caractéristique 2, Israel J. Math. 129 (2002) 317–361.

[19] A. Laghribi, On the generic splitting of quadratic forms in characteristic 2, Math. Z. 240 (2002) 711-730.

[20] A. Laghribi, P. Mammone, Isotropie d’une forme quadratique sur le corps des fonctions d’une quadrique en caractéristique 2, Bull. Belg. Math. Soc. 9 (2002) 167–176.

[21] P. Mammone, D. Shapiro, The Albert quadratic form for an algebra of degree four, Proc. Am. Math. Soc. 105 (1989) 525-530.

[22] P. Mammone, J.-P. Tignol, A. Wadsworth, Fields of characteristic 2 with prescribed u-invariant, Math. Ann. 290 (1991) 109–128.

[23] A. Merkurjev, Rost’s degree formula, Notes of a short course in Lens, June 2001.

[24] D. Orlov, A. Vishik, V. Voevodsky, Motivic cohomology of Pfister quadrics and Milnor’s conjecture on quadratic forms, preprint 1998.

[25] M. Rost, On quadratic forms isotropic over the function field of a conic, Math. Ann. 288 (1990), 511–513.

[26] C.-H. Sah, Symmetric bilinear forms and quadratic forms, J. Algebra 20 (1972) 144–160.