WITT KERNELS OF QUADRATIC FORMS FOR MULTIQUADRATIC EXTENSIONS IN CHARACTERISTIC 2

DETLEV W. HOFFMANN

Abstract. Let F be a field of characteristic 2 and let K/F be a purely inseparable extension of exponent 1. We show that the extension is excellent for quadratic forms. Using the excellence we recover and extend results by Aravire and Laghribi who computed generators for the kernel W_{q}(K/F) of the natural restriction map W_{q}(F) → W_{q}(K) between the Witt groups of quadratic forms of F and K, respectively, where K/F is a finite multiquadratic extension of separability degree at most 2.

1. Introduction

Throughout this article, we will only consider fields of characteristic 2. Let K be a finite multiquadratic extension of a field F of separability degree at most 2, in other words, K = F (\sqrt{a_{1}}, …, \sqrt{a_{n}}), or K = F (\sqrt{a_{1}}, …, \sqrt{a_{n}}, ℘^{−1}(b)), a_{i}, b ∈ F^{∗}, where ℘^{−1}(b) is a root of X^{2} + X + b. In [1], Aravire and Laghribi computed the kernel W_{q}(K/F) of the natural map (induced by scalar extension) W_{q}(F) → W_{q}(K) between the Witt groups of nonsingular quadratic forms over F and K, respectively. They show that

W_{q}(K/F) = ∑_{i=1}^{n} ⟨1, a_{i}⟩_{b} ⊗ W_{q}(F) ,

in the purely inseparable case, and

W_{q}(K/F) = W (F) ⊗ [1, b] + ∑_{i=1}^{n} ⟨1, a_{i}⟩_{b} ⊗ W_{q}(F)

in the case of separability degree 2, where ⟨1, a_{i}⟩_{b} is understood to be a binary bilinear form in its diagonal notation, [a, b] represents the nonsingular quadratic form ax^{2} + xy + by^{2}, and W_{q}(F) is considered as a module over the Witt ring of nonsingular symmetric bilinear forms W (F). The proof of Aravire and Laghribi uses differential forms. Actually, they prove more, namely they determine the kernel of the restriction map in Kato’s cohomology H_{2}^{n}(F) → H_{2}^{n}(K) and then deduce the result on the Witt kernels by using Kato’s theorem [7] that H_{2}^{∗}(F) is naturally isomorphic to the graded Witt module of quadratic forms over F. For more details, we refer the reader to Aravire and Laghribi’s article [1] and the references there.

The purpose of the present paper is to give a new more elementary proof of these results on the Witt kernels. Our approach is completely different. On the one hand, we show less since we don’t get the results on the kernels for the graded Witt modules. On the other hand, we show more. Recall that an extension K/F is called excellent (for quadratic forms) if for any quadratic form q over F, the anisotropic part of q over K is defined over F, i.e. there exists an anisotropic form φ over F with (q_{K})_{an} ≅ φ_{K}. We will show that the extension K/F is excellent if it is purely inseparable of exponent 1, i.e. if K^{2} ⊆ F. Using this, we can determine the Witt kernel for the compositum of such an exponent 1 extension with the function field of a quadratic or bilinear Pfister form. Note that the case K = F (\sqrt{a_{1}}, ⋯, \sqrt{a_{n}}, ℘^{−1}(b)) can be interpreted as the compositum of a finite exponent 1 extension with the function field of a 1-fold quadratic Pfister form, thus it also becomes a special case of our result.

2010 Mathematics Subject Classification. Primary 11E04; Secondary 11E81 12F15.

Key words and phrases. Quadratic form, bilinear form, Pfister form, Witt group, excellent extension, purely inseparable extension, exponent of an inseparable extension.

The research on this paper has been supported in part by the DFG Projekt “Annihilators and kernels in Kato’s cohomology in positive characteristic and in Witt groups in characteristic 2”.

Our main result is the following:

Main Theorem. Let K/F be a field extension such that K^{2} ⊂ F.

(i) The extension K/F is excellent for quadratic forms.

(ii)

W_{q}(K/F) = ∑_{t∈K^{∗2}} ⟨1, t⟩_{b} ⊗ W_{q}(F) .

In the next section, we introduce some basic terminology on bilinear and quadratic forms in characteristic 2. In the third section, we prove the main theorem, and in the fourth section, we provide some remarks concerning our main result and we generalize and extend it.

2. Terminology and definitions

For all undefined terminology on quadratic and bilinear forms, in particular in characteristic 2, we refer to [2] and [6]. We assume throughout that F is a field of characteristic 2. Bilinear forms are always assumed to be symmetric, and underlying vector spaces of bilinear and quadratic forms are always finite-dimensional.

Let (B, V) be a bilinear form over an F -vector space V. The radical is defined to be rad(B) = {x ∈ V | B(x, V) = 0}, and B is said to be nonsingular if rad(B) = 0. We define the value sets D_{F} (B) = {B(x, x) | x ∈ V \ {0}}, D_{F}^{0} (B) = D_{F} (B) ∪ {0} and D_{F}^{∗} (B) = D_{F} (B)∩F^{∗}. B is called isotropic if D_{F} (B) = D_{F}^{0} (B) and anisotropic if D_{F} (B) = D_{F}^{∗} (B). One has the usual notions of isometry ≅, orthogonal sum ⊥ and tensor product ⊗ of bilinear forms. In the sequel, we always assume bilinear forms to be nonsingular.

A 2-dimensional isotropic bilinear form is called a metabolic plane, in which case one can always find a basis such that the Gram matrix with respect to that basis is of the shape

( \begin{matrix}0 & 1 \\ 1 & a \end{matrix} )

for some a ∈ F. If a = 0, this is called a hyperbolic plane. A bilinear form B is said to be metabolic (resp. hyperbolic) if it is the orthogonal sum of metabolic (resp. hyperbolic) planes. It is not difficult to see that a form B is hyperbolic iff D_{F} (B) = {0}.

If the Gram matrix of a form B with respect to a certain basis is a diagonal matrix with entries a_{i}, 1 ≤ i ≤ n = dim(B), then we write B ≅ ⟨a_{1}, ⋯, a_{n}⟩_{b}. A diagonalization exists iff B is not hyperbolic.

A bilinear form B can be decomposed as B ≅ B_{an} ⊥ B_{m} with B_{an} anisotropic and B_{m} metabolic. B_{an} is uniquely determined up to isometry, but generally not B_{m}.

We call two bilinear forms B and B′ Witt equivalent if B_{an} ≅ B′_{an}. The equivalence classes together with addition induced by ⊥ and multiplication induced by ⊗ define the Witt ring of F denoted by W (F).

An n-fold bilinear Pfister form (or bilinear n-Pfister for short) is a form of type ⟨⟨a_{1}, …, a_{n}⟩⟩_{b} := ⟨1, a_{1}⟩_{b} ⊗ … ⊗ ⟨1, a_{n}⟩_{b} for some a_{i} ∈ F^{∗}. A bilinear Pfister form π is round, i.e., π ≅ xπ for all x ∈ D_{F} (π)^{∗}, and π is isotropic iff π is metabolic.

Now let (q, V) be a quadratic form over an F -vector space V, with associated bilinear form B_{q}(x, y) = q(x + y) + q(x) + q(y). One defines the radical rad(q) = rad(B_{q}) and calls q nonsingular if rad(q) = 0 and totally singular if rad(q) = V. Isometry ≅ and orthogonal sum ⊥ are defined in the usual way, and we define D_{F} (q) = {q(x) | x ∈ V \ {0}}, D_{F}^{0} (q) = D_{F} (q) ∪ {0} and D_{F}^{∗} (q) = D_{F} (q) ∩ F^{∗}, and we call q isotropic resp. anisotropic if D_{F} (q) = D_{F}^{0} (q) resp. D_{F} (q) = D_{F}^{∗} (q). A nonsingular quadratic form can always be written with respect to a suitable basis as ax^{2}+xy+by^{2} with a, b ∈ F, and we denote this form by [a, b]. Totally singular forms are exactly the diagonal forms a_{1}x_{1}^{2} +…+a_{n}x_{n}^{2}, for which we write ⟨a_{1}, …, a_{n}⟩. A hyperbolic plane ℍ is a 2-dimensional nonsingular isotropic quadratic form and one has ℍ ≅ [0, 0], and a quadratic form is called hyperbolic if it is an orthogonal sum of hyperbolic planes. The form [a, b] is hyperbolic iff ab ∈ ℘(F) = {c^{2} + c | c ∈ F }.

Each quadratic form q has a decomposition q ≅ q_{r} ⊥ q_{s} with q_{r} nonsingular and q_{s} totally singular. In fact, q_{s} is nothing but the restriction of q to rad(q) and it is thus uniquely determined, whereas q_{r} is generally not uniquely determined up to isometry, but dim(q_{r}) is uniquely determined. This decomposition can be refined as follows:

q ≅ \underbrace{ℍ ⊥ … ⊥ ℍ}_{k times} ⊥ q_{0} ⊥ ⟨a_{1}, …, a_{m}⟩ ⊥ ⟨\underbrace{0, …, 0}_{ℓ times}⟩

such that q_{0} is nonsingular and q_{an} = q_{0} ⊥ ⟨a_{1}, …, a_{m}⟩ is anisotropic. In this decomposition, k is uniquely determined and called the Witt index i_{W} (q) of q, ℓ is uniquely determined and called the defect i_{d}(q) of q, q_{an} is uniquely determined up to isometry and called the anisotropic part of q, and (k × ℍ) ⊥ q_{an} is also uniquely determined up to isometry and called the nondefective part of q.

If (q, V) and (q′, V′) are quadratic forms defined on F -vector spaces V and V′, respectively, then we say that q′ dominates q, q ≺ q′, it there exists an injective F -linear map t : V → V′ with q′(tx) = q(x) for all x ∈ V.

We call two quadratic forms q, q′ Witt equivalent, q ∼ q′, if q_{an} ≅ q′_{an}. The classes of nonsingular quadratic forms together with addition induced by the orthogonal sum form the Witt group W_{q}(F) of quadratic forms. W_{q}(F) has a natural structure as W (F)-module, essentially given by scaling ⟨a⟩_{b} ⊗ q = aq.

An n-fold quadratic Pfister form (or quadratic n-Pfister for short) is a form of type ⟨⟨a_{1}, …, a_{n}]] := ⟨⟨a_{1}, …, a_{n−1}⟩⟩_{b} ⊗ [1, a_{n}] for some a_{1}, …, a_{n−1} ∈ F^{∗}, a_{n} ∈ F. A quadratic Pfister form π is round, i.e., π ≅ xπ for all x ∈ D_{F} (π)^{∗}, and π is isotropic iff π is hyperbolic.

If φ is a (quadratic or bilinear) form over F and if K/F is a field extension, then we write φ_{K} := φ ⊗ K for the form obtained by scalar extension. This induces a natural homomorphism W (F) → W (K) resp. W_{q}(F) → W_{q}(K) whose kernel will be denoted by W (K/F) resp. W_{q}(K/F). In analogy to the definition of excellence of field extensions in the theory of quadratic forms in characteristic ≠ 2 as defined in [3], we say that K/F is excellent for quadratic resp. bilinear forms if for any quadratic resp. bilinear form φ over F there exists a form ψ over F with (φ_{K})_{an} ≅ ψ_{K}, in other words, the anisotropic part of φ over K is defined over F.

3. Witt kernels and excellence for extensions of exponent one

Let us now turn to the proof of the main theorem. Throughout this section, K/F will be a field extension with K^{2} ⊂ F. We define

J_{K/F} = ∑_{t∈K^{∗2}} ⟨1, t⟩_{b} ⊗ W_{q}(F) .

Remark 3.1. In [5], it was shown that a purely inseparable exponent 1 extension K/F is excellent for bilinear forms and that the bilinear Witt kernel W (K/F) is generated by {⟨1, t⟩_{b} | t ∈ K^{∗2}}. Hence, one readily has J_{K/F} = W (K/F) ⊗ W_{q}(F).

Lemma 3.2. J_{K/F} ⊂ W_{q}(K/F).

Proof. If t ∈ K^{∗2}, then clearly (⟨1, t⟩_{b})_{K} ≅ (⟨1, 1⟩_{b})_{K} is metabolic and hence, for any q ∈ ⟨1, t⟩_{b} ⊗ W_{q}(F) we have q_{K} = 0 ∈ W_{q}(K). □

Lemma 3.3. (Cf. [6, Lemma 2.2].) Let q ≅ q_{r} ⊥ q_{s} be a quadratic form over F with q_{r} nonsingular and q_{s} ≅ ⟨a_{1}, …, a_{n}⟩ totally singular, not all a_{i} = 0, and let K/F be any field extension. Then there exist 1 ≤ r ≤ n and 1 ≤ i_{1} < i_{2} < … < i_{t} ≤ n such that for q′_{s} ≅ ⟨a_{i_{1}}, …, a_{i_{r}}⟩, one has ((q_{s})_{K}))_{an} ≅ (q′_{s})_{K}. and (q_{K})_{an} ≅ φ ⊥ (q′_{s})_{K} for some nonsingular φ over K. In particular, every field extension is excellent for totally singular quadratic forms.

Let us restate the main theorem in more detail.

Theorem 3.4. Let q ≅ q_{r} ⊥ q_{s} be an anisotropic quadratic form over F with q_{r} nonsingular and q_{s} totally singular. Then there exists a nonsingular form q′_{r} over F with dim q′_{r} ≤ dim q_{r}, a totally singular form q′_{s} ≺ q_{s} over F, and a form ψ ∈ J_{K/F} such that

(i) (q_{K})_{an} ≅ (q′_{r} ⊥ q′_{s})_{K}, and

(ii) q ⊥ ψ ∼ q′_{r} ⊥ q_{s}.

In particular, K/F is excellent and W_{q}(K/F) = J_{K/F}.

Proof. First note that the result on the Witt kernel follows by considering the case dim q_{s} = 0 and q_{K} ∼ 0 in which case dim q′_{r} = 0 as well and thus q ≅ ψ.

To prove (i) and (ii), write q ≅ q_{r} ⊥ q_{s} with q_{r} nonsingular of dimension 2m, and q_{s} totally singular of dimension ℓ, and let n = dim q = 2m + ℓ.

If q_{K} is anisotropic, there is nothing to show. Also, one readily sees that by Lemma 3.3, we may assume that if dim q_{s} > 0, then (q_{s})_{K} is anisotropic.

So let us assume that q_{K} is isotropic, i.e. dim(q_{K})_{an} < n, and (q_{s})_{K} is anisotropic. In particular, m ≥ 1. If m = 1 and ℓ = 0, then q ≅ a[1, b] for some b ∉ ℘(F) and some a ∈ F^{∗}, and q_{K} being isotropic means that a[1, b]_{K} ≅ ℍ and hence b ∈ ℘(K) and thus F (℘^{−1}(b)) ⊂ K, a contradiction to K/F being purely inseparable.

Hence, we may assume m ≥ 1 and n = 2m + ℓ ≥ 3.

To prove the theorem, it suffices to construct a form β ∈ J_{K/F} such that for q̃ = (q ⊥ β)_{an} one has dim q̃ < dim q. Then we know that q_{K} ∼ q̃_{K} and we conclude by a simple induction on m.

Let us write

q ≅ \underbrace{[a_{1}, b_{1}] ⊥ … ⊥ [a_{m}, b_{m}]}_{q_{r}} ⊥ \underbrace{⟨c_{1}, …, c_{ℓ}⟩}_{q_{s}} .

Since q_{K} is isotropic, there exist x_{i}, y_{i}, z_{j} ∈ K, 1 ≤ i ≤ m, 1 ≤ j ≤ ℓ, with

(3.1)  0 = ∑_{i=1}^{m}(a_{i}x_{i}^{2} + x_{i}y_{i} + b_{i}y_{i}^{2}) + ∑_{j=1}^{ℓ} c_{j}z_{j}^{2}

and not all x_{i}, y_{i}, z_{j} equal to 0.

Suppose there exists i ∈ {1, …, m} with x_{i} = y_{i} = 0, or j ∈ {1, …, ℓ} with z_{j} = 0. Then write q ≅ q̂ ⊥ [a_{i}, b_{i}] resp. q ≅ q̂ ⊥ ⟨c_{i}⟩. Thus, dim q̂ < dim q and q̂_{K} is isotropic. We then may proceed by working with q̂ rather than q.

So we may assume that for each 1 ≤ i ≤ m we have x_{i} ≠ 0 or y_{i} ≠ 0, and that for each 1 ≤ j ≤ ℓ we have z_{j} ≠ 0. Without loss of generality, there exists 1 ≤ k ≤ m with x_{i} ≠ 0 for 1 ≤ i ≤ k, and x_{k+1} = … = x_{m} = 0. In particular, y_{i} ≠ 0 for k + 1 ≤ i ≤ m.

Note that K^{2} ⊆ F, so x_{i}^{2}, y_{i}^{2}, z_{j}^{2} ∈ F, so, if x_{i} ≠ 0, π_{i} ≅ ⟨1, x_{i}^{2}⟩_{b} ⊗ [a_{i}, b_{i}] ∈ J_{K/F} and [a_{i}, b_{i}] ⊥ π_{i} ∼ [a_{i}x_{i}^{2}, \frac{b_{i}}{x_{i}^{2}} ]. Put

q_{1} := [a_{1}x_{1}^{2}, \frac{b_{1}}{x_{1}^{2}} ] ⊥ … ⊥ [a_{k}x_{k}^{2}, \frac{b_{k}}{x_{k}^{2}} ] ⊥ [a_{k+1}, b_{k+1}] ⊥ … ⊥ [a_{m}, b_{m}] ⊥ q_{s} .

Then q_{1} ∼ q ⊥ π_{1} ⊥ … ⊥ π_{k}, dim q = dim q_{1} and q_{K} ∼ (q_{1})_{K}. Since we are interested in the anisotropic part of q over K and since q_{1} differs from q by a form in J_{K/F} ⊆ W_{q}(K/F), and since

a_{i}x_{i}^{2} + x_{i}y_{i} + b_{i}y_{i}^{2} = (a_{i}x_{i}^{2}) · 1^{2} + 1 · (x_{i}y_{i}) + \frac{b_{i}}{x_{i}^{2}} (x_{i}y_{i})^{2} ,

we may assume furthermore, by replacing q by q_{1}, that in Eq. 3.1 and by reassigning the letters a_{i}, b_{i}, we have x_{1} = … = x_{k} = 1, so together with the other assumptions

0 = ∑_{i=1}^{k}(a_{i} + y_{i} + b_{i}y_{i}^{2}) + ∑_{i=k+1}^{m} b_{i}y_{i}^{2} + ∑_{j=1}^{ℓ} c_{j}z_{j}^{2} .

Now using repeatedly [t, u] ⊥ [v, w] ≅ [t + v, u] ⊥ [v, u + w], we get

(3.2) [a_{1}, b_{1}] ⊥ … ⊥ [a_{k}, b_{k}] ≅ [a_{1}+…+a_{k}, b_{1}] ⊥ [a_{2}, b_{1}+b_{2}] ⊥ … ⊥ [a_{k}, b_{1}+b_{k}] .

Note that furthermore

(3.3) \begin{aligned}0 & = & (a_{1} + … + a_{k}) + (y_{1} + … + y_{k}) + b_{1}(y_{1} + … + y_{k})^{2} \\ & & +(b_{1} + b_{2})y_{2}^{2} + … + (b_{1} + b_{k})y_{k}^{2} + ∑_{i=k+1}^{m} b_{i}y_{i}^{2} + ∑_{j=1}^{ℓ} c_{j}z_{j}^{2} , \end{aligned}

so in view of Eqs. 3.2 and 3.3, we may assume k = 1, i.e. in Eq. 3.1, we have x_{1} = 1, x_{2} = … = x_{m} = 0. We also still may assume that y_{i} ≠ 0 for 2 ≤ i ≤ m or else we could again omit those terms [a_{i}, b_{i}] with i ≥ 2 and y_{i} = 0. By adding forms in J_{K/F} similarly as above, we may furthermore assume y_{2} = … = y_{m} = 1, so that Eq. 3.1 becomes

(3.4)  0 = a_{1} + y_{1} + b_{1}y_{1}^{2} + ∑_{i=1}^{k} b_{i} + ∑_{j=1}^{ℓ} c_{j}z_{j}^{2} .

Note that, again similarly as before, (3.5) [a_{2}, b_{2}] ⊥ … ⊥ [a_{m}, b_{m}] ≅ [a_{2}, b_{2} + … + b_{m}] ⊥ [a_{2} + a_{3}, b_{3}] ⊥ … ⊥ [a_{2} + a_{m}, b_{m}]

Eqs. 3.4 and 3.5 together imply that in Eq. 3.1, we may assume m ≤ 2, x_{1} = 1 and y_{2} = 1 (if m = 2), still with 2m + ℓ ≥ 3.

Suppose ℓ = 0. Then m = 2 and we are in the case q ≅ [a_{1}, b_{1}] ⊥ [a_{2}, b_{2}] and there exists y ∈ K with a_{1} + y + b_{1}y^{2} + b_{2} = 0. But y^{2} ∈ F, therefore also y ∈ F and we have that q is also isotropic over F, a contradiction.

Suppose now ℓ > 0 and m = 2. So q ≅ [a_{1}, b_{1}] ⊥ [a_{2}, b_{2}] ⊥ ⟨c_{1}, … c_{ℓ}⟩ and Eq. 3.1 becomes with the above assumptions 0 = a_{1} + y + b_{1}y^{2} + b_{2} + c_{1}z_{1}^{2} + … + c_{ℓ}z_{ℓ}^{2} with y, z_{i} ∈ K. Recall that by assumption all z_{i} ∈ K^{∗}, hence z_{i}^{2} ∈ F^{∗}. Then put

τ_{i} := ⟨1, z_{i}^{2}⟩_{b} ⊗ [a_{2}z_{i}^{2}, c_{i}] ≅ [a_{2}z_{i}^{2}, c_{i}] ⊥ [c_{i}z_{i}^{2}, a_{2}] ∈ J_{K/F}

and we have

[a_{2}, d] ⊥ ⟨c_{i}⟩ ⊥ τ_{i} ∼ [a_{2}, d + c_{i}z_{i}^{2}] ⊥ ⟨c_{i}⟩

and hence, with τ ≅ τ_{1} ⊥ … ⊥ τ_{ℓ} ∈ J_{K/F}, we get

[a_{2}, b_{2}] ⊥ q_{s} ⊥ τ ∼ [a_{2}, b_{2} + c_{1}z_{1}^{2} + … + c_{ℓ}z_{ℓ}^{2}] ⊥ q_{s} .

Put b′_{2} = b_{2} + c_{1}z_{1}^{2} + … + c_{ℓ}z_{ℓ}^{2}. This shows that for q_{2} := [a_{1}, b_{1}] ⊥ [a_{2}, b′_{2}] ⊥ q_{s} we have q ⊥ τ ∼ q_{2}, hence q_{K} ∼ (q_{2})_{K} and (with the same y as before) a_{1} + y + b_{1}y^{2} + b′_{2} = 0, so q_{2} is isotropic, and q̃ := (q_{2})_{an} is the desired form with dim q̃ < dim q_{2} = dim q.

Suppose finally that ℓ > 0 and m = 1, so q ≅ [a_{1}, b_{1}] ⊥ ⟨c_{1}, … c_{ℓ}⟩ and Eq. 3.1 becomes with the above assumptions 0 = a_{1} + y + b_{1}y^{2} + c_{1}z_{1}^{2} + … + c_{ℓ}z_{ℓ}^{2} with y, z_{i} ∈ K.

If y = 0, put

ρ_{i} := ⟨1, z_{i}^{2}⟩_{b} ⊗ [b_{1}z_{i}^{2}, c_{i}] ≅ [b_{1}z_{i}^{2}, c_{i}] ⊥ [b_{1}, c_{i}z_{i}^{2}] ∈ J_{K/F} ,

and we have

[d, b_{1}] ⊥ ⟨c_{i}⟩ ⊥ ρ_{i} ∼ [d + c_{i}z_{i}^{2}, b_{1}] ⊥ ⟨c_{i}⟩ ,

and with ρ := ρ_{1} ⊥ … ⊥ ρ_{ℓ} ∈ J_{K/F}, we get similarly as before

q ⊥ ρ ∼ [\underbrace{a_{1} + c_{1}z_{1}^{2} + … + c_{ℓ}z_{ℓ}^{2}}_{0}, b_{1}] ⊥ q_{s} ∼ q_{s} .

and q̃:= q_{s} is the desired form.

If y ≠ 0, put

ν_{i} := ⟨1, \frac{z_{i}^{2}}{y^{2}} ⟩_{b} ⊗ [a_{1} \frac{z_{i}^{2}}{y^{2}} , c_{i}] ≅ [a\frac{z_{i}^{2}}{y^{2}} , c_{i}] ⊥ [a_{1}, c_{i} \frac{z_{i}^{2}}{y^{2}} ] ∈ J_{K/F} ,

and we have

[a_{1}, d] ⊥ ⟨c_{i}⟩ ⊥ ν_{i} ∼ [a_{1}, d + c_{i} \frac{z_{i}^{2}}{y^{2}} ] ⊥ ⟨c_{i}⟩ ,

and with ν := ν_{1} ⊥ … ⊥ ν_{ℓ} ∈ J_{K/F} and b′_{1} := b_{1} + c_{1} \frac{z_{1}^{2}}{y^{2}} + … + c_{ℓ} \frac{z_{ℓ}^{2}}{y^{2}} we get

q ⊥ ν ∼ [a_{1}, b′_{1}] ⊥ q_{s}

and (with the same y as before) a_{1} +y +b′_{1}y^{2} = 0, i.e. [a_{1}, b′_{1}] is isotropic and hence hyperbolic, and q ⊥ ν ∼ q_{s}, and again q̃:= q_{s} is the desired form. This completes the proof. □

4. Some corollaries and remarks

We are now interested in extensions that are obtained by composing purely inseparable exponent 1 extensions with function fields of Pfister forms, and in the Witt kernels of these new extensions.

If π is an n-fold quadratic Pfister form over F (n ≥ 1), then let F (π) be the function field of the projective quadric X_{π} given by the equation π = 0.

If B = ⟨⟨a_{1}, …, a_{n}⟩⟩_{b}, a_{i} ∈ F^{∗}, is an n-fold bilinear Pfister form with associated totally singular quadratic form q(x) = B(x, x), i.e. q = ⟨⟨a_{1}, …, a_{n}⟩⟩, then there exist k ≤ n and 1 ≤ i_{1} < … < i_{k} ≤ n with q_{an} ≅ ⟨⟨a_{i_{1}}, …, a_{i_{k}}⟩⟩ (see, e.g., [6, §8]). Put B_{0} = ⟨⟨a_{i_{1}}, …, a_{i_{k}}⟩⟩_{b}. We define the function field F (B) := F (q). Then F (q) is a purely transcendental extension of F (q_{an}) = F (B_{0}) (see, e.g., [4, Remark 7.4(iii)]). Since anisotropic forms stay anisotropic over purely transcendental extensions, we have that W_{q}(F (B)/F) = W_{q}(F (B_{0})/F). Note that if k = 0, i.e. q_{an} ≅ ⟨1⟩ and thus B_{0} ≅ ⟨1⟩_{b}, we have F (B_{0}) = F and hence W_{q}(F (B)/F) = W_{q}(F (B_{0})/F) = 0.

We have the following results on Witt kernels for function fields of Pfister forms.

Proposition 4.1. Let n ≥ 1 and let π is an n-fold quadratic Pfister form over F, or let B = ⟨⟨a_{1}, …, a_{n}⟩⟩_{b}, a_{i} ∈ F^{∗}, be an n-fold bilinear Pfister form with associated totally singular quadratic form q(x) = B(x, x) such that dim q_{an} ≥ 2, i.e. [F^{2}(a_{1}, …, a_{n}) : F^{2}] > 1.

(i) ([8, Th. 1.4(2)] or [2, Cor. 23.6].) Let φ ∈ W_{q}(F (π)/F) be anisotropic, then there exists a bilinear form β with φ ≅ β ⊗π. In particular, W_{q}(F (π)/F) = W (F) ⊗ π.

(ii) ([8, Th. 1.4(3)].) Let φ be an anisotropic form in W_{q}(F (B)/F), then there exists a nonsingular quadratic form τ with φ ≅ B_{0} ⊗ τ. In particular, W_{q}(F (B)/F) = B_{0} ⊗ W_{q}(F).

Remark 4.2. Note that for b ∈ F \℘(F), π = [1, b] is an anisotropic 1-fold quadratic Pfister form and F (π) = F (℘^{−1}(b)), and for a ∈ F \F^{2}, B = ⟨1, a⟩_{b} is an anisotropic 1-fold bilinear Pfister form and F (B) = F (\sqrt{a}), so the kernels in Proposition 4.1 are nothing but the well known ones for separable resp. inseparable quadratic extensions.

Corollary 4.3. Let n ≥ 1 and let π be an n-fold quadratic Pfister form over F. Let K be a purely inseparable extension of F of exponent 1. Let L = K(π). Then W_{q}(L/F) = W_{q}(K/F) + W F ⊗ π.

Proof. Clearly, W_{q}(K/F) + W F ⊗ π ⊆ W_{q}(L/F). Conversely, let φ be a form over F with φ ∈ W_{q}(L/F). By the excellence property of K/F, there exists a form ψ over F with (φ_{K})_{an} ≅ ψ_{K}. If dim ψ = 0 then φ ∈ W_{q}(K/F) and we are done. So suppose dim ψ > 0. Since 0 = φ_{L} = ψ_{L} ∈ W_{q}(L), it follows by Proposition 4.1 that there exists a bilinear form β over L with ψ_{L} ≅ β ⊗ π_{L}. By the excellence of K/F, we may in fact assume that β is already defined over F, see [3, Prop. 2.11] (the argument there works also in characteristic 2), so we may put ψ ≅ β ⊗ π. But then φ − β ⊗ π ∈ W_{q}(K/F) and hence φ ∈ W_{q}(K/F) + W F ⊗ π as desired. □

Lemma 4.4. Let K/F be a purely inseparable extension of exponent 1, let B be a bilinear Pfister form over F and let ρ be a nonsingular quadratic form over K such that B_{K} ⊗ ρ is defined over F, i.e. there exists a quadratic form φ over F with φ_{K} ≅ B_{K} ⊗ ρ. Then there exists a nonsingular quadratic form ρ_{0} over F with φ_{K} ≅ B_{K} ⊗ ρ ≅ (B ⊗ ρ_{0})_{K}.

Proof. This is essentially [9, Th. 1], except that there K was assumed to be an inseparable quadratic extension. But the proof works in exactly the same way by using the fact that for any b ∈ K, one has that [1, b] is defined over F since [1, b] ≅ [1, b^{2}] over K with b^{2} ∈ F since K^{2} ⊂ F. □

Corollary 4.5. Let K be a purely inseparable extension of F of exponent 1. Let n ≥ 1 and let B = ⟨⟨a_{1}, …, a_{n}⟩⟩_{b}, a_{i} ∈ F^{∗}, be an n-fold bilinear Pfister form over F with associated totally singular quadratic form q = ⟨⟨a_{1}, …, a_{n}⟩⟩, and let k ≤ n and 1 ≤ i_{1} < … < i_{k} ≤ n be such that (q_{K})_{an} ≅ ⟨⟨a_{i_{1}}, …, a_{i_{k}}⟩⟩_{K}. Put B_{0} = ⟨⟨a_{i_{1}}, …, a_{i_{k}}⟩⟩_{b}. Let L = K(B). If dim B_{0} = 1 then W_{q}(L/F) = W_{q}(K/F). If dim B_{0} > 1 then W_{q}(L/F) = W_{q}(K/F) + B_{0} ⊗ W_{q}(F).

Proof. By the remarks preceding Proposition 4.1, we have that K(B)/K(B_{0}) is purely transcendental, hence W_{q}(L/F) = W_{q}(K(B_{0})/F). We are done if dim B_{0} = 1 as then K(B_{0}) = K. So let us assume dim B_{0} > 1. Clearly, W_{q}(K/F) + B_{0} ⊗ W_{q}(F) ⊆ W_{q}(L/F). Conversely, let φ be a form over F with φ ∈ W_{q}(L/F) = W_{q}(K(B_{0})/F). By the excellence property of K/F, there exists a form ψ over F with (φ_{K})_{an} ≅ ψ_{K}. If dim ψ = 0 then φ ∈ W_{q}(K/F) and we are done. So suppose dim ψ > 0. Since 0 = φ_{K(B_{0})} = ψ_{K(B_{0})} ∈ W_{q}(K(B_{0})), it follows from Proposition 4.1 that there exists a nonsingular form τ over K with ψ_{K} ≅ (B_{0})_{K} ⊗ τ. By Lemma 4.4, we may assume that τ is already defined over F, hence we may put ψ ≅ B_{0}⊗τ. But then φ−B_{0}⊗τ ∈ W_{q}(K/F) and hence φ ∈ W_{q}(K/F)+B_{0}⊗W_{q}(F) as desired. □

Theorem 3.4 essentially says that W_{q}(K/F) is additively generated by forms ⟨1, t⟩_{b} ⊗ [c, d] with t ∈ K^{∗2} and c, d ∈ F. We next show that we don’t need all t ∈ K^{∗2} for this to be true. Recall that a subset T ⊂ F is called 2-independent if for all finite subsets {t_{1}, …, t_{n}} ⊆ T with n ∈ ℕ and t_{i} ≠ t_{j} for i ≠ j, one has

[F (\sqrt{t_{1}}, …, \sqrt{t_{n}}) : F] = [F^{2}(t_{1}, …, t_{n}) : F^{2}] = 2^{n} .

A 2-independent set T with F^{2}(T) = F is called a 2-basis of F.

If K/F is a purely inseparable exponent 1 extension, then clearly there exists a 2-independent set T ⊆ F with K = F (\sqrt{t} | t ∈ T).

Corollary 4.6. Let K/F be a purely inseparable exponent 1 extension, and let T ⊆ F be a 2-independent set such that K = F (\sqrt{t} | t ∈ T). Then

W_{q}(K/F) = ∑_{t∈T} ⟨1, t⟩_{b} ⊗ W_{q}(F) .

Proof. Obviously, ∑_{t∈T} ⟨1, t⟩_{b} ⊗ W_{q}(F) ⊆ W_{q}(K/F) as t ∈ T implies t ∈ K^{∗2}.

For the reverse inclusion, let q ∈ W_{q}(K/F) be anisotropic. It is clear that there exists already a finite subset T′ ⊂ T such that for K′ = F (\sqrt{t} | t ∈ T′), we have q ∈ W_{q}(K′/F). By invoking Proposition 4.1 and Remark 4.2 together with Corollary 4.5, an easy induction on the cardinality |T′| shows that W_{q}(K′/F) = ∑_{t∈T′} ⟨1, t⟩_{b}⊗W_{q}(F). Hence q ∈ W_{q}(K′/F) ⊆ ∑_{t∈T} ⟨1, t⟩_{b}⊗W_{q}(F) as desired. □

Corollary 4.6 together with Corollary 4.3 (in the case n = 1, see Remark 4.2) immediately imply the result by Aravire and Laghribi [1] mentioned in the introduction.

Corollary 4.7. Let K = F (\sqrt{a_{1}}, …, \sqrt{a_{n}}), a_{i} ∈ F^{∗}, and L = K(℘^{−1}(b)) for some b ∈ F. Then

\begin{aligned}W_{q}(K/F) & = & ∑_{i=1}^{n} ⟨1, a_{i}⟩_{b} ⊗ W_{q}(F) \\ W_{q}(L/F) & = & W (F) ⊗ [1, b] + ∑_{i=1}^{n} ⟨1, a_{i}⟩_{b} ⊗ W_{q}(F) \end{aligned}

Remark 4.8. A different way of deriving the previous corollary for W_{q}(K/F) from our Theorem 3.4 is as follows. It suffices to show that each form of type ⟨1, t⟩_{b}⊗[c, d], t ∈ K^{∗2}, c, d ∈ F, can be written in W_{q}(F) as a sum of forms of type ⟨1, a_{i}⟩_{b} ⊗ q_{i} for suitable nonsingular quadratic forms q_{i} over F.

Now t ∈ F^{2}(a_{1}, …, a_{n}), so for each I ⊆ {1, …, n} there exist x_{I} ∈ F such that t = ∑_{I⊂{1,...,n}}(∏_{i∈I} a_{i})x_{I}^{2}. The desired result now follows by a straightforward induction on n using the following relations in W_{q}(F), where we assume w ∈ F, u, v, x ∈ F^{∗} with u + v ≠ 0 in the third relation:

• ⟨1, 1⟩_{b} ⊗ [1, w] = 0;

• ⟨1, ux^{2}⟩_{b} ⊗ [1, w] = ⟨1, u⟩_{b} ⊗ [1, w];

• ⟨1, u + v⟩_{b} ⊗ [1, w] = ⟨1, u⟩_{b} ⊗ [1, \frac{wu}{u+v}] + ⟨1, v⟩_{b} ⊗ [1, \frac{wv}{u+v}];

• ⟨1, uv⟩_{b} ⊗ [1, w] = ⟨1, u⟩_{b} ⊗ [1, w] + ⟨1, v⟩_{b} ⊗ [u, \frac{w}{u}].

References

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Fakultät für Mathematik, Technische Universität Dortmund, 44221 Dortmund, Germany

E-mail address: detlev.hoffmann@math.tu-dortmund.de