Isotropy of Quadratic Spaces

in Finite and Infinite Dimension

Karim Johannes Becher, Detlev W. Hoffmann

Received: October 6, 2006 Revised: September 13, 2007

Communicated by Alexander Merkurjev

Abstract. In the late 1970s, Herbert Gross asked whether there exist fields admitting anisotropic quadratic spaces of arbitrarily large finite dimensions but none of infinite dimension. We construct examples of such fields and also discuss related problems in the theory of central simple algebras and in Milnor K-theory.

2000 Mathematics Subject Classification: 11E04, 11E81, 12D15, 12E15, 12F20, 12G05, 12G10, 16K20, 19D45 Keywords and Phrases: quadratic form, isotropy, infinite-dimensional quadratic space, u-invariant, function field of a quadric, totally indefinite form, real field, division algebra, quaternion algebra, symbol algebra, Galois cohomology, cohomological dimension, Milnor K-theory

1 Introduction

A quadratic space over a field F is a pair (V, q) of a vector space V over F together with a map q : V ⟶ F such that

• q(λx) = λ^{2}q(x) for all λ ∈ F, x ∈ V, and

• the map b_{q} : V × V ⟶ F defined by b_{q}(x, y) = q(x + y) − q(x) − q(y) (x, y ∈ V) is F -bilinear.

If dim V = n < ∞, one may identify (after fixing a basis of V) the quadratic space (V, q) with a form (a homogeneous polynomial) of degree 2 in n variables. Via this identification, a finite-dimensional quadratic space over F will also be referred to as quadratic form over F. Recall that a quadratic space (V, q) is said to be isotropic, if there exists x ∈ V \ {0} such that q(x) = 0; otherwise, (V, q) is said to be anisotropic. Questions about isotropy are at the core of the algebraic theory of quadratic forms over fields. A natural and much studied field invariant in this context is the so-called u-invariant of a field F. If F is of characteristic not 2 and nonreal (i.e. −1 is a sum of squares in F), then u(F) is defined to be the supremum of the dimensions of anisotropic finite-dimensional quadratic forms over F. See Section 2 for the general definition of the u-invariant. The main purpose of the present article is to give examples of fields having infinite u-invariant but not admitting any anisotropic infinite-dimensional quadratic space. Assume now that the quadratic space (V, q) over F is anisotropic. For any positive integer n ≤ dim(V), let V_{n} be any n-dimensional subspace of V and consider the restriction q_{n} = q|_{V_{n}}. Clearly, the n-dimensional quadratic form (V_{n}, q_{n}) is again anisotropic. This simple argument shows that if there is an anisotropic quadratic space over F of infinite dimension, then there exist anisotropic quadratic forms over F of dimension n for all n ∈ ℕ. While this observation is rather trivial, it motivates us to examine the converse statement. If we assume that the field F has anisotropic quadratic forms of arbitrarily large finite dimensions, does this imply the existence of some anisotropic quadratic space (V, q) over F of infinite dimension? As already mentioned, this is generally not so. It appears that originally this question has been formulated by Herbert Gross. He concludes the introduction to his book ‘Quadratic forms in infinitedimensional vector spaces’ [12] (published in 1979) by the following sample of ‘a number of pretty and unsolved problems’ in this area, which we state in his words (cf. [12], p. 3):

1.1 Question (Gross). Is there any commutative field which admits no anisotropic ℵ_{0}-form but which has infinite u-invariant, i.e. admits, for each n ∈ ℕ, some anisotropic form in n variables?

Note that implicitly, Gross is looking for a nonreal field, because anisotropic quadratic spaces of infinite dimension always exist over real fields. (We use the term ‘real field’ for what is often called ‘formally real field’.) Indeed, one observes that the field F is real if and only if the infinite-dimensional quadratic space (V, q) given by V = F^{(ℕ)} and q : V ⟶ F, (x_{i}) ⟼ ∑ x_{i}^{2} is anisotropic. By restricting to those quadratic spaces that are totally indefinite, i.e. indefinite with respect to every field ordering, one can formulate a meaningful analogue of the Gross Question also for real fields, to which we will provide a solution as well. We also study the Gross Question in characteristic 2 where one has to distinguish between bilinear forms and quadratic forms. For quadratic forms, one furthermore has to distinguish the cases of nonsingular quadratic forms and of arbitrary quadratic forms. The analogue to the Gross Question for nonsingular quadratic forms in characteristic 2 can be treated in more or less the same way as in characteristic not 2, simply by invoking suitable analogues of the results that we use in our proofs in the case of characteristic different from 2. Yet, if translated to bilinear forms or to arbitrary quadratic forms (possibly singular) in characteristic 2, it is not difficult to show that the Gross Question has in fact a negative answer, in other words, the ‘bilinear’ resp. ‘general quadratic’ u-invariant is infinite if and only if there exist infinite-dimensional anisotropic bilinear resp. quadratic spaces.

The paper is structured as follows. In the next section, we are going to discuss in more detail the u-invariant of a field and some related concepts. In Section 3 we will give two different constructions of nonreal fields, each giving a positive answer to the Gross Question. All our constructions will be based on Merkurjev’s method where one starts with an arbitrary field and then uses iterated extensions obtained by composing function fields of quadrics to produce an extension with the desired properties. Our first construction will show the following:

1.2 Theorem I. Let F be a field of characteristic different from 2. There exists a field extension K/F with the following properties:

(i) K has no finite extensions of odd degree.

(ii) For any binary quadratic form β over K, there is an upper bound on the dimensions of anisotropic quadratic forms over K that contain β.

(iii) For any k ∈ ℕ, there is an anisotropic k-fold Pfister form over K.

In particular, K is a perfect, nonreal field of infinite u-invariant, I^{k}K ≠ 0 for all k ∈ ℕ, and any infinite-dimensional quadratic space over K is isotropic.

Here and in the sequel, I^{k}F stands for the k^{th} power of IF, the fundamental ideal consisting of classes of even-dimensional forms in the Witt ring W F of F. The proof of this theorem only uses some basic properties of Pfister forms and standard techniques from the theory of function fields of quadratic forms. Varying this construction and using this time products of quaternion algebras and Merkurjev’s index reduction criterion (see [24] or [38], Théorème 1), we will then show the following:

1.3 Theorem II. Let F be a field of characteristic different from 2. There exists a field extension K/F with the following properties:

(i) K has no finite extensions of odd degree and I^{3}K = 0.

(ii) For any binary quadratic form β over K, there is an upper bound on the dimensions of anisotropic quadratic forms over K that contain β.

(iii) For any k ∈ ℕ, there is a central division algebra over K that is decomposable into a tensor product of k quaternion algebras.

In particular, K is a nonreal field of infinite u-invariant, and any infinitedimensional quadratic space over K is isotropic. Furthermore, K is perfect and of cohomological dimension 2.

In Section 4, we will show two analogous theorems for real fields.

1.4 Theorem III. Assume that F is real. Then there exists a field extension K/F with the following properties:

(i) K has a unique ordering.

(ii) K has no finite extensions of odd degree and I^{3}K is torsion free.

(iii) For any totally indefinite quadratic form β over K, there is an upper bound on the dimensions of anisotropic quadratic forms over K that contain β.

(iv) For any k ∈ ℕ, there is a central division algebra over K that is decomposable into a tensor product of k quaternion algebras.

In particular, K is a real field of infinite u-invariant, and any totally indefinite quadratic space of infinite dimension over K is isotropic; moreover, the cohomological dimension of K(\sqrt{−1}) is 2.

While this can be seen as a counterpart to Theorem II for real fields, we can also prove an analogue of Theorem I in this situation.

1.5 Theorem IV. Assume that F is real. Then there exists a field extension K/F with the following properties:

(i) K has a unique ordering.

(ii) K has no finite extensions of odd degree.

(iii) For any totally indefinite quadratic form β over K, there is an upper bound on the dimensions of anisotropic quadratic forms over K that contain β.

(iv) for any k ∈ ℕ, there is an element a ∈ K^{×} which is a sum of squares in K, but not a sum of k squares.

In particular, K is a real field for which the Pythagoras number, the Hasse number, and the u-invariant are all infinite, the torsion part of I^{k}K is nonzero for all k ∈ ℕ, and any totally indefinite quadratic space of infinite dimension over K is isotropic.

In Section 5, we will discuss the Gross Question for quadratic, nonsingular quadratic, and symmetric bilinear forms in characteristic 2. As already mentioned, for nonsingular quadratic forms, we obtain similar results as in characteristic different from 2, whereas for arbitrary quadratic forms and for symmetric bilinear forms the answer turns out to be negative.

In the final Section 6, we discuss an abstract version of the Gross Question, formulated for an arbitrary monoïd together with two subsets satisfying some requirements. We give examples of such monoïds whose elements are well known objects associated to an arbitrary field, such as central simple algebras or symbols in Milnor K-theory modulo a prime p. In some of the cases that we shall discuss, the answer to (the analogue of) the Gross Question will be positive, in others it will be negative.

For all prerequisites from quadratic form theory in characteristic different from 2 needed in the sequel, we refer to the books of Lam and Scharlau (see [20], [21] and [34]). In general, we use the standard notations introduced there. However, we use a different sign convention for Pfister forms: Given a_{1}, …, a_{r} ∈ F^{×}, we write ⟨⟨a_{1}, …, a_{r}⟩⟩ for the r-fold Pfister form ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{r}⟩. If φ is a quadratic form over F and n ∈ ℕ, we denote by n × φ the n-fold orthogonal sum φ ⊥ ⋯ ⊥ φ. A quadratic space (V, q) is said to be nonsingular if the radical

Rad(V, q) = {x ∈ V | b_{q}(x, y) = 0 for all y ∈ V }

is reduced to 0. Anisotropic quadratic spaces in characteristic different from 2 are obviously always nonsingular, but this need not be so in characteristic 2. Given two quadratic spaces (resp. forms) φ and ψ over F. We say that ψ is a subspace (resp. subform) of φ if ψ is isometric to the restriction of φ to some subspace of the underlying vector space of φ. We write ψ ⊂ φ if there exists a quadratic space τ over F such that φ ≅ ψ ⊥ τ. If φ, ψ are quadratic forms over F with ψ nonsingular, then ψ ⊂ φ if and only if ψ is a subform of φ. Unless stated otherwise, the terms ‘form’ or ‘quadratic form’ will always stand for ‘nonsingular quadratic form’. A binary form is a 2-dimensional quadratic form. We recall the definition of the function field F (φ) associated to a nonsingular quadratic form φ over F in characteristic different from 2. If dim(φ) ≥ 3 or if dim(φ) = 2 and φ is anisotropic, then F (φ) is the function field of the projective quadric given by the equation φ = 0. We put F (φ) = F if φ is the hyperbolic plane or if dim(φ) ≤ 1. We refer to [34], Chapter 4, §5, or [21], Chapter X, for the crucial properties of function field extensions. They will play a prominent rôle in all our constructions. Let K/F be an arbitrary field extension. If φ is a quadratic form over F, then we denote by φ_{K} the quadratic form over K obtained by scalar extension from F to K. Similarly, given an F -algebra A, we write A_{K} for the K-algebra A⊗_{F} K. Central simple algebras are by definition finite-dimensional. A central simple algebra without zero-divisors will be called a ‘division algebra’ for short. For the basics about central simple algebras and the Brauer group of a field, the reader is referred to [34], Chapter 8, or [31], Chapters 12-13.

2 The derived u-invariant

In this section, all fields are assumed to be of characteristic different from 2.

The question about the existence of an anisotropic infinite-dimensional quadratic space over the field F can be rephrased within the framework of finitedimensional quadratic form theory, as we shall see now. We call a sequence of quadratic forms (φ_{n})_{n∈ℕ} over F a chain of quadratic forms over F if, for any n ∈ ℕ, we have dim(φ_{n}) = n and φ_{n} ⊂ φ_{n+1}. Given such a chain (φ_{n})_{n∈ℕ} over F, the direct limit over the quadratic spaces φ_{n} with the appropriate inclusions has itself a natural structure of a nonsingular quadratic space over F of dimension ℵ_{0} (countably infinite). We denote this quadratic space over F by lim_{n∈ℕ}(φ_{n}) and observe that it is anisotropic if and only if φ_{n} is anisotropic for all n ∈ ℕ. Moreover, any infinite-dimensional nonsingular quadratic space over F contains a subspace isometric to the direct limit lim_{n∈ℕ}(φ_{n}) for some chain (φ_{n})_{n∈ℕ} and we thus get:

2.1 Proposition. There exists an anisotropic quadratic space of infinite dimension over F if and only if there exists a chain of anisotropic quadratic forms (φ_{n})_{n∈ℕ} over F.

Recall that a form φ is torsion if its Witt class is a torsion element in the Witt ring W F. In [9], Elman and Lam defined the u-invariant of F as

u(F) = sup {dim(φ) | φ is an anisotropic torsion form over F } .

It is well known that if F is nonreal, then any form over F is torsion, in which case the above supremum is actually taken over all anisotropic forms over F. If F is real, then Pfister’s Local-Global Principle says that torsion forms are exactly those forms that have signature zero with respect to each ordering of F (i.e. that are hyperbolic over each real closure of F). In the remainder of this section, we are mainly concerned with nonreal fields.

It will be convenient to consider also the following relative u-invariants. Given an anisotropic quadratic form φ over F, we define

u(φ, F) = sup {dim(ψ) | ψ anisotropic form over F with φ ⊂ ψ} .

Note that, trivially, dim(φ) ≤ u(φ, F). If F is nonreal, we further have that u(φ, F) ≤ u(F) with equality if dim φ = 1. Moreover, if φ_{1} and φ_{2} are anisotropic forms over F such that φ_{1} ⊂ φ_{2}, then u(φ_{1}, F) ≥ u(φ_{2}, F). We introduce now the derived u-invariant of F as

u′(F) = sup {dim(φ) | φ anisotropic form over F with u(φ, F) = ∞}.

Whenever there exists an anisotropic form φ over F with u(φ, F) = ∞, we have u′(F) > 0; if no such forms exist, we put u′(F) = sup ∅ = 0.

2.2 Proposition. If there exists an infinite-dimensional quadratic space over F, then u′(F) = ∞.

Proof. Assume that there exists an anisotropic infinite-dimensional quadratic space over F. Then there is also a chain (φ_{n})_{n∈ℕ} of anisotropic forms over F. Obviously, u(φ_{n}, F) = ∞ for any n ∈ ℕ, and therefore u′(F) = ∞.

In particular, the proposition shows that u′(F) = ∞ if F is a real field. Certainly, one could modify the definition of u′ to make this invariant more interesting for real fields, but we will not pursue this matter here.

2.3 Proposition. Assume that F is nonreal. Then u(F) is finite if and only if u′(F) = 0.

Proof. If u(F) = ∞, then u(⟨1⟩, F) = u(F) = ∞ and thus u′(F) ≥ 1. On the other hand, if u(F) < ∞, then there is no anisotropic form φ over F such that u(φ, F) = ∞, and therefore u′(F) = 0.

By the previous two propositions, any nonreal field F with 0 < u′(F) < ∞ will yield an example that answers the Gross Question in the positive. Now Theorem I and Theorem II each say that nonreal fields K with u′(K) = 1 do exist.

2.4 Lemma. For the field F ((t)) of Laurent series in the variable t over F, one has

u′(F ((t)) ) = 2 u′(F) .

The proof of this lemma is straightforward and based on the well known relationship between quadratic forms over F and over F ((t)) (see [20], Chapter VI, Proposition 1.9). Details are left to the reader.

2.5 Corollary. Let m ∈ ℕ. Then there exists a nonreal field L such that u′(L) = 2^{m}. Moreover, L can be constructed such that in addition I^{m+3}L = 0, or I^{r}L ≠ 0 for all r ∈ ℕ, respectively.

Proof. Theorem I and Theorem II, respectively, assert the existence of such fields for m = 0. The induction step from m to m + 1 is clear from the preceding lemma.

This raises the following question.

2.6 Question. Does there exist a nonreal field F with u′(F) = ∞ such that every infinite-dimensional quadratic space over F is isotropic?

3 Nonreal fields with infinite u-invariant

We are going to give a construction, in several variants, which allows us to prove the theorems formulated in the introduction. The proof that the field obtained by this construction has infinite u-invariant will be based on known facts about the preservation of properties such as anisotropy of a fixed quadratic form, or absence of zero-divisors in a central simple algebra, under certain types of field extensions.

First, we consider a finite field extension K/F of odd degree. Springer’s Theorem (see [20], Chapter VII, Theorem 2.3) says that any anisotropic quadratic form over F stays anisotropic after scalar extension from F to K.

Springer’s Theorem has an analogue in the theory of central simple algebras. It says that if D is a (central) division algebra over F with exponent equal to a power of 2 and if K/F is a finite field extension of odd degree, then the K-algebra D_{K} = D ⊗_{F} K is also a division algebra (see [31], Section 13.4, Proposition (vi)). Both statements also hold in characteristic 2. One can immediately generalise them to ‘odd’ algebraic extensions that are not necessarily finite. An algebraic extension L/F is called an odd closure of F if L is F -isomorphic to M^{G}, where M is an algebraic (resp. separable) closure of F if char(F) ≠ 2 (resp. char(F) = 2), and G is a 2-Sylow subgroup of the Galois group of M/F. Then L itself has no odd degree extension and all finite subextensions of F inside L are of odd degree. In particular, L is perfect if char(F) ≠ 2. We call a field extension K/F an odd extension if it can be embedded into an odd closure of F. In this case, K/F is algebraic, thus equal to the direct limit of its finite subextensions, which are all of odd degree. We thus get immediately the following (where we do not make any assumption on the characteristic).

3.1 Lemma. Let K/F be an odd extension.

(i) Any anisotropic quadratic form over F stays anisotropic over K.

(ii) Any central division algebra of exponent 2 over F remains a division algebra over K.

For the remainder of this section, all fields are assumed to be of characteristic different from 2. We now consider extensions of the type F (φ)/F, where F (φ) is the function field of a quadratic form φ over F.

3.2 Lemma. Let π be an anisotropic Pfister form over F and φ a form over F with dim(φ) > dim(π). Then π stays anisotropic over F (φ).

Proof. By the assumption on the dimensions, φ is certainly not similar to any subform of π. Therefore, by [34], Theorem 4.5.4 (ii), π_{F (φ)} is not hyperbolic. Hence π_{F (φ)} is anisotropic as it is a Pfister form (see [34], Lemma 2.10.4).

3.3 Remark. The statement of the last lemma is actually a special case of a more general phenomenon. Let φ and π be anisotropic forms over F such that, for some n ∈ ℕ, one has dim(π) ≤ 2^{n} < dim(φ). Then π stays anisotropic over F (φ) (see [14]). In the particular situation where π is an n-fold Pfister form, we immediately recover (3.2).

The next statement was the key in Merkurjev’s construction of fields of arbitrary even u-invariant (see [24]). It is readily derived from [38], Théorème 1.

3.4 Theorem (Merkurjev). Let D be a division algebra over F of exponent 2 and degree 2^{m}, where m > 0. Let φ be a quadratic form over F such that dim(φ) > 2m + 2 or φ ∈ I^{3}F. Then D_{F (φ)} is a division algebra.

3.5 Remark. It is also well known that if K/F is a purely transcendental extension, then anisotropic forms (resp. division algebras) over F stay anisotropic (resp. division) over K. We will use this fact repeatedly, especially when K = F (φ) is the function field of an isotropic quadratic form φ over F, in which case F (φ)/F is purely transcendental of transcendence degree dim(φ) − 2 (see [34], 4.5.2 (vi)).

3.6 Proof of Theorem I.

Recall that F is an arbitrary field of characteristic different from 2. We define recursively a tower of fields (F_{n})_{n∈ℕ}, starting with F_{0} = F. Suppose that for a certain n ≥ 1 the field F_{n−1} has already been defined. Let F_{n−1}^{#} be an odd closure of F_{n−1} and let

F_{n−1}^{(n)} = F_{n−1}^{#}(X_{1}^{(n)}, …, X_{n}^{(n)})

where X_{1}^{(n)}, …, X_{n}^{(n)} are indeterminates over F_{n−1}^{#}. We define F_{n} as the free compositum^{1} of all function fields F_{n−1}^{(n)}(φ) where φ ranges over all anisotropic forms defined over F_{n−1} such that, for some j < n, dim(φ) = 2^{j} + 1 and φ contains a binary form defined over F_{j}. Let K be the direct limit of the tower of fields (F_{n})_{n∈ℕ}. We are going to show that the field K has the following properties:

(i) K has no finite extensions of odd degree.

(ii) For any binary quadratic form β over K, there is an upper bound on the dimensions of anisotropic quadratic forms over K that contain β.

(iii) For any k ∈ ℕ, there is an anisotropic k-fold Pfister form over K.

Once these are established the remaining claims in Theorem I will follow. Indeed, (ii) implies that every infinite-dimensional quadratic space over K is isotropic and that K is nonreal, whereas (iii) implies that u(K) = ∞ and that I^{k}K ≠ 0 for all k ∈ ℕ. Finally, since char(K) = char(F) ≠ 2, it follows from (i) that K is perfect.

(i) Consider an irreducible polynomial f over K of odd degree. Then f is defined over F_{n} for some n ∈ ℕ. Since K contains F_{n+1} which in turn contains an odd closure of F_{n}, it follows that f has degree one. This shows that K is equal to its odd closure. (ii) Consider an anisotropic binary form β over K. There is some j ∈ ℕ such that β is defined over F_{j}. Let φ be a form of dimension 2^{j} + 1 over K containing β. Let n > j be an integer such that φ is defined over F_{n−1}. Then by construction, F_{n} contains F_{n−1}^{(n)}(φ) and φ is therefore isotropic over F_{n} and thus over K. This shows that u(β, K) ≤ 2^{j}. Here, j depends on the binary form β, but in any case we have that u(β, K) is finite, proving (ii). (iii) Given positive integers n and j, we write F_{n,j} for the compositum of F_{n} with the algebraic closure of F_{j} inside a fixed algebraic closure of K. Similarly, we write F_{n−1,j}^{#} and F_{n−1,j}^{(n)}, for the compositum of F_{n−1}^{#}, F_{n−1}^{(n)}, respectively, with the algebraic closure of F_{j}.

1See [21], p. 333, for a precise description of the notion of ‘free compositum’ of a family of function fields of quadratic forms.

Documenta Mathematica 12 (2007) 473–504

[figure: F_{n,j}; F_{n}; compositum; F_{n−1,j}^{(n)}; F_{n−1}^{(n)}; purely transc.; F_{n−1,j}^{#}; F_{n−1}^{#}; odd; F_{n−1,j}; F_{n−1}]

From now on, let n > j. Note that F_{n−1,j}^{(n)} = F_{n−1,j}^{#}(X_{1}^{(n)}, …, X_{n}^{(n)}) is a purely transcendental extension of F_{n−1,j}^{#}. Further, F_{n−1,j}^{#} is an odd extension of F_{n−1,j}. Using (3.1), it follows that every anisotropic form over F_{n−1,j} stays anisotropic over F_{n−1,j}^{#}, hence also over F_{n−1,j}^{(n)}. Moreover, F_{n,j} is obtained from F_{n−1,j}^{(n)} as a free compositum of certain function fields F_{n−1,j}^{(n)}(φ) where φ is a form defined over the subfield F_{n−1} of F_{n−1,j}^{(n)}, either dim(φ) ≥ 2^{j+1} + 1, or dim(φ) = 2^{ℓ} + 1 with ℓ ≤ j in which case φ contains a binary subform defined over F_{ℓ} ⊂ F_{j}. But in this latter case, φ is isotropic over F_{n−1,j}^{(n)} and thus F_{n−1,j}^{(n)}(φ)/F_{n−1,j}^{(n)} is purely transcendental.

Consider now an anisotropic m-fold Pfister form π defined over F_{n−1,j}^{(n)}, where m ≤ j + 1. Since, by the above, F_{n,j} is obtained from F_{n−1,j}^{(n)} as a compositum of function fields of forms of dimension at least 2^{j+1} + 1 and of purely transcendental extensions, (3.2) and (3.5) imply that π stays anisotropic over F_{n,j}. But then π stays anisotropic over F_{n,j}^{(n+1)} as well. Repeating this, we see that π stays anisotropic over all the fields F_{m,j} for all m ≥ n. Let now k be any positive integer. Let π denote the k-fold Pfister form ⟨⟨X_{1}^{(k)}, …, X_{k}^{(k)}⟩⟩. This form is defined over F_{k−1}^{(k)}. Since X_{1}^{(k)}, …, X_{k}^{(k)} are algebraically independent over F_{k−1}, hence also over its algebraic closure F_{k−1,k−1} = F_{k−1,k−1}^{#}, we know that π is still anisotropic when considered

as a form over the field F_{k−1,k−1}^{(k)} = F_{k−1,k−1}^{#}(X_{1}^{(k)}, …, X_{n}^{(k)}). Now the above argument shows that, for any n ≥ k, the form π is anisotropic over F_{n,k−1} and, thus, over F_{n}. This implies that π is anisotropic over K, the direct limit of the fields F_{n}. Hence we showed that for any k ∈ ℕ, there exists an anisotropic k-fold Pfister form over K.

3.7 Proof of Theorem II.

Again, we define recursively a tower of fields (F_{n})_{n∈ℕ}, starting with F_{0} = F. Suppose that for a certain n ≥ 1, the field F_{n−1} is defined. As before, let F_{n−1}^{#} denote an odd closure of F_{n−1}. This time we define

F_{n−1}^{(n)} = F_{n−1}^{#}( X_{1}^{(n)}, Y_{1}^{(n)}, …, X_{n}^{(n)}, Y_{n}^{(n)})

where X_{1}^{(n)}, Y_{1}^{(n)}, …, X_{n}^{(n)}, Y_{n}^{(n)} are indeterminates over F_{n−1}^{#}. Let F_{n} denote the free compositum of the function fields F_{n−1}^{(n)}(φ) where φ is an anisotropic form over F_{n−1} such that

• φ is a 3-fold Pfister form, or

• dim(φ) = 2j + 3 for some j < n and φ contains a binary subform defined over F_{j}.

Let K be the direct limit of the tower of fields (F_{n})_{n∈ℕ}. We want to show that K has the following properties:

(i) K has no finite extensions of odd degree and I^{3}K = 0.

(ii) For any binary quadratic form β over K, there is an upper bound on the dimensions of anisotropic quadratic forms over K which contain β.

(iii) For any k ∈ ℕ, there is a central division algebra over K that is decomposable into a tensor product of k quaternion algebras.

Note that (iii) implies that u(K) = ∞ (see [24] or [28], Lemma 1.1(d)), while (ii) prohibits the existence of infinite-dimensional anisotropic quadratic spaces over K. Now the field K is perfect and nonreal by (i). Furthermore, (i) and (iii) together imply that the cohomological dimension of K is exactly 2 (see [24]).

(i) As in the proof of Theorem I, we see that K has no finite extensions of odd degree. Let π be an arbitrary 3-fold Pfister form over K. It is defined as a 3-fold Pfister form over F_{n−1} for some n ≥ 1. By the construction of the field F_{n}, π becomes isotropic over F_{n} and thus over K. Hence, every 3-fold Pfister form over K is isotropic and therefore hyperbolic. Since I^{3}K is additively generated by the 3-fold Pfister forms over K (see [34], p. 156), we conclude that I^{3}K = 0. (ii) Let β be an anisotropic binary form over K. There is an integer j ∈ ℕ such that β is defined over F_{j}. Let φ be any form of dimension 2j + 3 over K containing β. There is some integer n > j such that φ is defined over F_{n−1}. Since F_{n−1}^{#}(φ) is part of the compositum F_{n}, φ becomes isotropic over F_{n} and thus over K. Therefore u(β, K) ≤ 2j + 2, establishing (ii). (iii) For positive integers n and j, we denote by F_{n,j}, F_{n−1,j}^{#}, F_{n−1,j}^{(n)} the composita of the fields F_{n}, F_{n−1}^{#}, F_{n−1}^{(n)}, respectively, with the algebraic closure of F_{j} inside a fixed algebraic closure of K. Assume from now on that n > j. Similarly as in the proof of Theorem I, we have that F_{n−1,j}^{(n)} is equal to F_{n−1,j}^{#}(X_{1}^{(n)}, Y_{1}^{(n)}, …, X_{n}^{(n)}, Y_{n}^{(n)}), a purely transcendental extension of F_{n−1,j}^{#}, which in turn is an odd extension of F_{n−1,j}. Using (3.1) and (3.5), it follows that every division algebra of exponent 2 over F_{n−1,j} remains a division algebra after scalar extension to F_{n−1,j}^{(n)}.

Moreover, F_{n,j} is obtained from F_{n−1,j}^{(n)} as a free compositum of certain function fields F_{n−1,j}^{(n)}(φ) where φ is a form defined over F_{n−1,j}^{(n)} which is either a 3-fold Pfister form, or which has dimension at least 2j +3, or which contains a binary form defined over F_{j} and thus is isotropic over F_{n−1,j}^{(n)}. Hence, by (3.4) and (3.5), any division algebra over F_{n−1,j}^{(n)} of exponent 2 and of degree at most 2^{j} remains a division algebra after scalar extension to the field F_{n,j}. Consider now a central simple algebra D of exponent 2 and degree 2^{j} over F_{j−1}^{(j)} for some j ∈ ℕ. Assume that for some n > j, the algebra D will stay a division algebra after extending scalars to F_{n−1,j}^{(n)}. Combining the observations above, we see that D also remains a division algebra when we extend scalars to F_{n,j}, or even to F_{n,j}^{(n+1)}. Repeating this argument shows that D will stay a division algebra after scalar extension to F_{N−1,j}^{(N)} for any N ≥ n. Let now k be a positive integer and let D denote the tensor product of quaternion algebras (X_{1}^{(k)}, Y_{1}^{(k)}) ⊗ ⋯ ⊗ (X_{k}^{(k)}, Y_{k}^{(k)}) over the field F_{k−1}^{(k)}. This is a division algebra over F_{k−1}^{(k)} of degree 2^{k} and of exponent 2. Since X_{1}^{(k)}, Y_{1}^{(k)}, …, X_{k}^{(k)}, Y_{k}^{(k)} are algebraically independent over the field F_{k−1}, hence also over its algebraic closure F_{k−1,k−1} = F_{k−1,k−1}^{#}, it follows that

D_{F_{k−1,k−1}^{(k)}} is a division algebra over the field F_{k−1,k−1}^{(k)}. Now the argument above applies, showing that D_{F_{n,k−1}} is a division algebra over F_{n,k−1} for any n ≥ k. But then D_{F_{n}} is a division algebra for any n ≥ k, implying that the tensor product of k quaternion algebras D_{K} is a division algebra over K.

3.8 Remark. At first glance, it may seem that the fields K constructed in the proofs of the theorems are horrendously big. However, a closer inspection of the proofs reveals that if the field F we start with is infinite, the field K obtained by the construction will have the same cardinality as F. For example, if we start with F = ℚ, then the field K will be countable and thus can be embedded into ℂ.

4 Real fields and totally indefinite spaces

In our answer to the Gross Question, we had to construct a field F which in particular has the property that all infinite-dimensional quadratic spaces over F are isotropic. A real such field cannot exist as mentioned previously. In fact, for a quadratic space φ (of finite or infinite dimension) over a real field F to be isotropic, a necessary condition is that φ be totally indefinite, i.e. indefinite with respect to each ordering. To get a meaningful analogue to the Gross Question in the case of real fields, it is therefore reasonable to restrict our attention to quadratic spaces that are totally indefinite. We start this section with the definition of this notion and some general observations before proving the ‘real’ analogues to the constructions that answer the Gross Question.

Let F be real and let P be an ordering on F with corresponding order relation <_{P}. A quadratic space (V, q) over F is said to be indefinite at P, if there exist elements v_{1}, v_{2} ∈ V such that q(v_{1}) <_{P} 0 <_{P} q(v_{2}). If (V, q) is indefinite at every ordering of F, then we say that (V, q) is totally indefinite. Note that this definition of (total) indefiniteness extends the common one for quadratic forms. (By definition, if F is nonreal, every form over F is totally indefinite.) The Hasse number ũ of F is defined by

ũ(F) = sup {dim(φ) | φ anisotropic, totally indefinite form over F } .

Since any nontrivial torsion form is obviously totally indefinite, one has u(F) ≤ ũ(F). On the other hand, there are examples of real fields F where u(F) < ∞ while ũ(F) = ∞. For a survey on the possible pairs of values (u(F), ũ(F)), we refer to [15]. Recall that the Pythagoras number p(F) of F is the least integer m ≥ 1 such that every sum of squares is a sum of m squares in F if such an m exists, otherwise p(F) = ∞. It is well known and not difficult to see that if p(F) = ∞, then also u(F) = ũ(F) = ∞, and if u(F) > 0 then p(F) ≤ u(F). The following observation is useful when dealing with infinite-dimensional totally indefinite quadratic spaces.

4.1 Proposition. Every totally indefinite quadratic space over a real field F contains a finite-dimensional, nonsingular, totally indefinite quadratic subspace.

Proof. Let (V, q) be a totally indefinite quadratic space over F. We may assume (V, q) nonsingular. If (V, q) is isotropic then it contains a hyperbolic plane which yields the desired subspace. Hence, we may assume that (V, q) is anisotropic. In particular, any subspace of (V, q) is nonsingular. After scaling we may furthermore assume that there exists a vector v_{0} ∈ V with q(v_{0}) = 1. Since (V, q) is totally indefinite, for each ordering P there exists a vector v_{P} ∈ V such that q(v_{P}) <_{P} 0. Recall that the set of all orderings of F, denoted by X_{F}, is a compact topological space that has as a subbasis the clopen sets

H(a) = { P ∈ X_{F} | a ∈ P }

(see [32], Theorem 6.5). We put a_{P} = q(v_{P}) for every P ∈ X_{F}. Clearly, P ∈ H(−a_{P}) and hence X_{F} = ⋃_{P ∈X_{F}} H(−a_{P}). The compactness of X_{F} thus yields that there are finitely many orderings P_{1}, …, P_{n} ∈ X_{F} such that

X_{F} = H(−a_{P_{1}}) ∪ ⋯ ∪ H(−a_{P_{n}}).

We put v_{i} = v_{P_{i}} for 1 ≤ i ≤ n. By the last equality, for each ordering P of F we have q(v_{i}) <_{P} 0 for at least one i ∈ {1, …, n}. Let W be the subspace of V generated by the vectors v_{0}, v_{1}, …, v_{n}. Then it follows that (W, q) is an anisotropic, finite-dimensional, totally indefinite subspace of (V, q).

Recall that any ordering P of F can be extended to the odd closure of F as well as to any purely transcendental extension of F. From [10], Theorem 3.5, Remark 3.6, we cite the following simple criterion for when an ordering can be extended to the function field of a given quadratic form.

4.2 Lemma. Let P be an ordering of F and let {φ_{i}} be any family of quadratic forms over F of dimension at least 2. Then P can be extended to the free compositum of the F (φ_{i}) if and only if each φ_{i} is indefinite at P.

We are now going to modify the constructions presented in the last section and prove the remaining two theorems formulated in the introduction.

4.3 Proof of Theorem III.

This time, starting with the real field F = F_{0} and any ordering P_{0} on it, we construct a tower of fields with orderings (F_{n}, P_{n})_{n∈ℕ}, where the ordering P_{n+1} on F_{n+1} extends the ordering P_{n} on F_{n} for all n. Suppose now that the pair (F_{n−1}, P_{n−1}) has been defined for a certain n ≥ 1. Let F_{n−1}^{#} denote an odd closure of F_{n−1} and let P_{n−1}^{#} be any ordering on F_{n−1}^{#} extending P_{n−1}. Let

F_{n−1}^{(n)} = F_{n−1}^{#}(X_{1}^{(n)}, Y_{1}^{(n)}, …, X_{n}^{(n)}, Y_{n}^{(n)})

where X_{1}^{(n)}, Y_{1}^{(n)}, …, X_{n}^{(n)}, Y_{n}^{(n)} are indeterminates over F_{n−1}^{#}. Let P_{n−1}^{(n)} be any ordering on F_{n−1}^{(n)} extending P_{n−1}^{#}. Let now F_{n} be the free compositum of the function fields F_{n−1}^{(n)}(φ) where φ is an anisotropic form over F_{n−1} such that

• φ is a 3-fold Pfister form and indefinite at P_{n−1}, or

• dim(φ) = 2j + 3 for some j < n, and φ contains a binary form defined over F_{j} and indefinite at P_{j}.

Note that considered as forms over F_{n−1}^{(n)} and by the construction of our orderings, all the above forms are in fact totally indefinite at P_{n−1}^{(n)}. By (4.2), the ordering P_{n−1}^{(n)} extends to an ordering P_{n} on F_{n}. In particular, F_{n} is a real field. Note that, for any 2-fold Pfister form ρ over F_{n−1} and any a ∈ F_{n−1}, at least one of the 3-fold Pfister forms ρ ⊗ ⟨⟨a⟩⟩ and ρ ⊗ ⟨⟨−a⟩⟩ is indefinite at P_{n−1} and thus becomes hyperbolic over F_{n} by the construction of this field. Let K be the direct limit of the tower of fields (F_{n})_{n∈ℕ}. We will show that K has the following properties:

(i) K has a unique ordering which is given by P = ⋃_{n∈ℕ} P_{n}.

(ii) K has no finite extensions of odd degree and I^{3}K is torsion free.

(iii) For any totally indefinite quadratic form β over K, there is an upper bound on the dimensions of anisotropic quadratic forms over K that contain β.

(iv) For any k ∈ ℕ, there is a central division algebra over K that is decomposable into a tensor product of k quaternion algebras.

Once these properties of K are established, the remaining claims in Theorem III are immediate consequences:

• K is a real field and by (iii) and (4.1), every infinite-dimensional anisotropic quadratic space over K is definite with respect to the unique ordering.

• (i) implies that K is SAP (see, e.g., [32], § 9, for the definition of and some facts about SAP), I^{3}K is torsion free, and (iv) implies that the symbol length λ(K) of K is infinite. (Recall that the symbol length λ(K) is the smallest m ∈ ℕ such that each central simple algebra of exponent 2 over K is Brauer equivalent to a tensor product of at most m quaternion algebras provided such an integer exists, otherwise λ(K) = ∞.) It follows from [15], Theorem 1.5, that u(K) = ∞.

• (i) and (ii) yield that the cohomological dimension of K(\sqrt{−1}) is at most 2. (iv) then implies that it is exactly 2.

We now proceed to the proof of (i)–(iv).

(i) Since all the fields F_{n} (n ∈ ℕ) are real, the same holds for K. It follows from what we observed during the construction above that, for any a ∈ K^{×}, one of the forms ⟨⟨−1, −1, a⟩⟩ and ⟨⟨−1, −1, −a⟩⟩ is hyperbolic over K, which means that either a or −a is a sum of four squares in K. This shows that K is uniquely ordered. It is clear that the unique ordering on K is given by ⋃_{n∈ℕ} P_{n}. (ii) There is no change — compared to the previous constructions — in the argument that K has no finite extensions of odd degree.

The torsion subgroup of I^{3}K is generated by those 3-fold Pfister forms over K that are torsion. Indeed, this is a general fact (see [2], Corollary 2.7) which, however, could be proven very easily in our particular situation where K is uniquely ordered. Let π be any torsion 3-fold Pfister form over K. Then π is defined as a 3-fold Pfister form over F_{n−1} for some n ≥ 1. Since the unique ordering on K extends the ordering P_{n−1} on F_{n−1}, it follows that π (considered as 3-fold Pfister form over F_{n−1}) is indefinite at P_{n−1}. The construction of F_{n} then yields that π becomes isotropic and hence hyperbolic over F_{n}. Therefore, π is hyperbolic over K. This shows that I^{3}K is torsion free. (iii) Since K has a unique ordering, every (totally) indefinite form over K contains an indefinite binary subform. Hence, (iii) needs only to be proven for binary indefinite forms β. The proof goes along the same lines as that of (ii) in Theorem II. (iv) This part is identical to the corresponding part (iii) in the proof of Theorem II.

4.4 Proof of Theorem IV.

Again, starting with the real field F = F_{0} and any ordering P_{0} on it, we define a tower of ordered fields (F_{n}, P_{n})_{n∈ℕ} where the ordering P_{n+1} on F_{n+1} extends the ordering P_{n} on F_{n} for all n. Suppose that for a certain n ≥ 1 the pair (F_{n−1}, P_{n−1}) is already defined. Let F_{n−1}^{#} be an odd closure of F_{n−1} and let F_{n−1}^{(n)} be the rational function field F_{n−1}^{#}(X^{(n)}). As before, P_{n−1} extends to some ordering P_{n−1}^{#} of F_{n−1}^{#} which in turn extends to an ordering P_{n−1}^{(n)} on F_{n−1}^{(n)} = F_{n−1}^{#}(X^{(n)}) at which X^{(n)} is positive. We define F_{n} to be the free compositum of all function fields F_{n−1}^{(n)}(φ) where φ is an anisotropic form defined over F_{n−1} such that, for some j < n, we have dim(φ) = 2^{j} + 1 and φ contains an binary form which is defined over F_{j} and indefinite at P_{j}. By (4.2), P_{n−1}^{(n)} extends to an ordering P_{n} of F_{n}. Let K be the direct limit of the tower (F_{n})_{n∈ℕ}. We are going to establish the following properties:

(i) K has a unique ordering which is given by P = ⋃_{n∈ℕ} P_{n}.

(ii) K has no finite extensions of odd degree.

(iii) For any totally indefinite quadratic form β over K, there is an upper bound on the dimensions of anisotropic quadratic forms over K which contain β.

(iv) for any k ∈ ℕ, there is an element a ∈ K^{×} which is a sum of squares in K, but not a sum of k squares.

Note that (iv) implies that the Pythagoras number of K is infinite, which in turn forces the Hasse number and the u-invariant of K to be infinite as well. As before, (iii) implies that every infinite-dimensional anisotropic quadratic space over K is definite with respect to the unique ordering of K.

(i) Since each F_{n} is real, so is the direct limit K. Consider an arbitrary element a ∈ K^{×}. Then a ∈ F_{n} for some n ∈ ℕ. Now either ⟨1, −a⟩ or ⟨1, a⟩ is indefinite at P_{n}. Therefore, by construction, either 2^{n} × ⟨1⟩ ⊥ ⟨−a⟩ or 2^{n} × ⟨1⟩ ⊥ ⟨a⟩ becomes isotropic over the field F_{n+1}. Hence, a or −a is a sum of 2^{n} squares in K. It readily follows that K has a unique ordering given by ⋃_{n∈ℕ} P_{n}. (ii) K is equal to its odd closure, by the same arguments as before. (iii) The argument here is the same as for (iii) in the last proof. (iv) We denote by F_{n−1,j}, F_{n−1,j}^{#}, and F_{n−1,j}^{(n)}, the composita of F_{n−1}, F_{n−1}^{#}, and F_{n−1}^{(n)}, respectively, with the real closure of F_{j} at the ordering P_{j}. Assume now that n > j. Then we observe as before that every anisotropic quadratic form defined over F_{n−1,j} stays anisotropic over F_{n−1,j}^{(n)}. Note that F_{n,j} is obtained from F_{n−1,j}^{(n)} as a compositum of function fields F_{n−1,j}^{(n)}(φ) where φ is a form defined over F_{n−1,j}^{(n)} which is either of dimension at least 2^{j+1} + 1, or which contains a binary form defined over F_{j} and indefinite at P_{j} and which is therefore isotropic over F_{n−1,j}^{(n)}. As in part (iii) of the proof of Theorem I, we conclude that if π is an anisotropic m-fold Pfister form over F_{n−1,j}^{(n)} with m ≤ j + 1, then π stays anisotropic over F_{n,j}. Let now k ∈ ℕ. Then the (k + 1)-fold Pfister form 2^{k} × ⟨⟨X^{(k)}⟩⟩ is defined over F_{k−1}^{(k)} and is still anisotropic over F_{k−1,k−1}^{(k)}. It follows now from the above arguments that this form stays anisotropic over F_{n,k−1}, for all n > k. In particular, 2^{k} × ⟨⟨X^{(k)}⟩⟩ is anisotropic over all fields F_{n} for n ≥ k, thus also over K. This shows that the element X^{(k)} is not a sum of 2^{k} squares in K. On the other hand, by the construction we have X^{(k)} ∈ P, so that X^{(k)} is a sum of squares in K, by (i).

5 Fields of characteristic 2

Throughout this section, all fields considered will be of characteristic 2. To translate the Gross Question into this setting, we have to take into account the different types of objects for which analogous problems might be formulated: quadratic, nonsingular quadratic, and symmetric bilinear spaces. We maintain the convention to use the term ‘form(s)’ for finite-dimensional spaces. For nonsingular quadratic forms we shall obtain analogues to Theorems I and II stated in the introduction, thus obtaining a positive answer to (the corresponding formulation of) the Gross Question in this case, too. On the other hand, for arbitrary quadratic forms as well as for symmetric bilinear forms, the corresponding answer turns out to be negative. In fact, this is relatively easy to prove, so we treat these types of forms first.

We refer the reader to [3], [30], or [16] for further details on notation, terminology and basic results on quadratic and bilinear forms in characteristic 2. Let (V, q) be a quadratic space over a field F of characteristic 2 and b_{q} : V ×V → F its associated bilinear form given by b_{q}(x, y) = q(x + y) + q(x) + q(y). Recall that the radical of (q, V) is the F -subspace

V^{⊥} = Rad(q, V) = {x ∈ V | b_{q}(x, y) = 0 for all y ∈ V } .

The quadratic space (V, q) is said to be

• nonsingular if V^{⊥} = 0;

• singular if V^{⊥} ≠ 0;

• totally singular if V^{⊥} = V.

If we write V = V_{0}⊕V^{⊥} and we put q_{0} = q|_{V_{0}} and q_{ts} = q|_{V^{⊥}}, then q ≅ q_{0} ⊥ q_{ts} with q_{0} nonsingular and q_{ts} totally singular. If we also have q ≅ φ_{0} ⊥ φ_{ts} with φ_{0} nonsingular and φ_{ts} totally singular, then q_{ts} ≅ φ_{ts} (any isometry maps radicals bijectively to radicals), but q_{0} and φ_{0} might not be isometric. Note that (V, q) is totally singular if and only if q(x+y) = q(x)+q(y) for all x, y ∈ V. For a, b ∈ F, the 2-dimensional quadratic form aX^{2} +XY +bY^{2} is nonsingular, and we will denote it by [a, b]. The hyperbolic plane is then the form ℍ = [0, 0] = XY. For a_{1}, …, a_{s} ∈ F, the s-dimensional quadratic form ∑_{i=1}^{s} a_{i}X_{i}^{2} is totally singular, and it will be denoted by ⟨a_{1}, …, a_{s}⟩. Let now q be a quadratic form over F and let n = dim(q). Then there exist r, s ∈ ℕ with 2r + s = n and a_{1}, b_{1}, …, a_{r}, b_{r}, c_{1}, …, c_{s} ∈ F such that

q ≅ [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{r}, b_{r}] ⊥ ⟨c_{1}, …, c_{s}⟩ ,

and we clearly have q_{ts} ≅ ⟨c_{1}, …, c_{s}⟩. In particular, nonsingular quadratic forms are always of even dimension.

There are two versions of the u-invariant in characteristic 2, referring to the different types of quadratic forms, denoted by u and û, respectively. They are defined as follows:

u(F) = sup{dim(q) | q anisotropic nonsingular quadratic form over F }
û(F) = sup{dim(q) | q anisotropic quadratic form over F }

Clearly, we have u(F) ≤ û(F), and u(F) is always even if finite. One could define corresponding u-invariants also for the classes of anisotropic symmetric bilinear forms, and of anisotropic totally singular quadratic forms, respectively, but (5.3) below will show that both suprema thus obtained coincide with [F : F^{2}], the degree of inseparability of F.

We will now concentrate for a moment on totally singular quadratic spaces, a case that is very easy to treat.

For a field F of characteristic 2 we fix an algebraic closure \overline{F} and put \sqrt{F} = \{x ∈ \overline{F} | x^{2} ∈ F \}. Note that \sqrt{F} /F is a purely inseparable algebraic field extension of degree [F : F^{2}]. Hence the squaring map sq : x ↦ x^{2} yields a quadratic map sq_{F} : \sqrt{F} → F over F, and the quadratic space (\sqrt{F} , sq_{F}) is clearly of dimension [F : F^{2}].

5.1 Proposition. Let F be a field of characteristic 2. The quadratic space (\sqrt{F} , sq_{F}) is anisotropic, totally singular, and of dimension [F : F^{2}]. Any anisotropic totally singular quadratic space over F is isometric to a subspace of (\sqrt{F} , sq_{F}).

Proof. The first part is obvious. Consider now a totally singular quadratic space (V, q) over F and assume that it is anisotropic. We define

ρ : V ⟶ \sqrt{F} , v ⟼ \sqrt{q(v)} .

Since q is totally singular, ρ is F -linear and we have sq_{F} ∘ ρ = q. Since furthermore q is anisotropic, ρ is injective and thus (V, q) is isometric to the subspace (ρ(V), sq_{F} |_{ρ(V)}) of (\sqrt{F} , sq_{F}).

We will now briefly look at symmetric bilinear spaces (V, b) over a field F of characteristic 2. A symmetric bilinear space (V, b) is said to be isotropic if there exists x ∈ V \ {0} such that b(x, x) = 0, anisotropic otherwise. In other words, (V, b) is anisotropic if and only if (V, q_{b}) is so, where q_{b} : V → F is the induced quadratic map defined by q_{b}(x) = b(x, x).

5.2 Lemma. Let F be a field of characteristic 2 and V an F -vector space. There exists an anisotropic symmetric bilinear map b : V × V → F if and only if there exists an anisotropic totally singular quadratic map q : V → F.

Proof. By definition, a symmetric bilinear map b : V × V → F is anisotropic if and only if the associated totally singular quadratic map q_{b} : V → F is so. Now, given an anisotropic totally singular quadratic map q : V → F, it is not difficult to construct a symmetric bilinear map b : V ×V → F such that q = q_{b}. In fact, picking some F -basis (e_{i})_{i∈I} of V, we can define b by b(e_{i}, e_{j}) = δ_{ij}q(e_{i}) for i, j ∈ I. All this implies the claim.

The previous two statements readily imply the following.

5.3 Corollary. Let F be a field of characteristic 2. Then [F : F^{2}] = ∞ if and only if there exist anisotropic totally singular quadratic spaces and anisotropic symmetric bilinear spaces of infinite dimension over F. Moreover, if [F : F^{2}] < ∞, then

[F : F^{2}] = sup {dim(q) | q anisotr. tot. singular quadratic form over F }
= sup {dim(b) | b anisotr. symmetric bilinear form over F }

We next consider general quadratic forms in characteristic 2 and the corresponding û-invariant. The second part of the following statement is [22], Corollary 1.

5.4 Proposition. Let F be a field of characteristic 2. Then û(F) < ∞ if and only if [F : F^{2}] < ∞, in which case

[F : F^{2}] ≤ û(F) ≤ 2[F : F^{2}] .

Proof. If [F : F^{2}] = ∞ then the last corollary readily implies that û(F) = ∞. Now suppose [F : F^{2}] < ∞. Then [F : F^{2}] ≤ û(F) also follows from the corollary. To prove the second inequality, consider an anisotropic quadratic form q over F, say,

q = [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{r}, b_{r}] ⊥ ⟨c_{1}, …, c_{s}⟩ , a_{1}, b_{1}, …, a_{r}, b_{r}, c_{1}, …, c_{s} ∈ F.

Then the totally singular subform ⟨a_{1}, …, a_{r}, c_{1}, …, c_{s}⟩ is anisotropic as well, hence r + s ≤ [F : F^{2}] and thus dim(q) ≤ 2 [F : F^{2}]. It follows that û(F) ≤ 2 [F : F^{2}].

So far we have shown in this section that the Gross Question (1.1) has actually a negative answer when it is reformulated for general quadratic forms, for totally singular quadratic forms, or for symmetric bilinear forms over a field of characteristic 2. Let us now return to the case of nonsingular quadratic forms and spaces. To motivate the Gross Question (1.1), we first shall show that the existence of an infinite-dimensional anisotropic nonsingular quadratic space implies the existence of such spaces in every finite even dimension. Again, for quadratic forms φ and ψ over F we write φ ⊂ ψ if there exists a quadratic form τ such that ψ ≅ φ ⊥ τ. It is clear that if any two of the quadratic forms φ, ψ, τ are nonsingular, then so is the third. We call a sequence of nonsingular quadratic forms (φ_{n})_{n∈ℕ} over F a chain of nonsingular quadratic forms over F if, for any n ∈ ℕ, we have dim(φ_{n}) = 2n and φ_{n} ⊂ φ_{n+1}. Note that we need even dimension for nonsingularity. Given such a chain (φ_{n})_{n∈ℕ} over F, the direct limit over the quadratic spaces φ_{n} with the appropriate inclusions is again a nonsingular quadratic space over F of countably infinite dimension. We denote this quadratic space over F by lim_{n∈ℕ}(φ_{n}) and observe that it is anisotropic if and only if φ_{n} is anisotropic for all n ∈ ℕ.

5.5 Lemma. Any infinite-dimensional nonsingular quadratic space over F has a subspace isometric to lim_{n∈ℕ}(φ_{n}) for some chain (φ_{n})_{n∈ℕ} of nonsingular quadratic forms.

Proof. Let (V, q) be nonsingular with dim(V) = ∞ and let b = b_{q}.

(i) Let x ∈ V \ {0}. The nonsingularity implies the existence of y ∈ V such that b(x, y) ≠ 0. Clearly, x and y are linearly independent as b(x, x) = 0. Let U_{1} ⊂ V be the subspace spanned by x and y. Let φ_{1} = q|_{U_{1}}. One readily sees that φ_{1} is nonsingular. (ii) If U ⊂ V is any finite-dimensional subspace with q|_{U} nonsingular, then the usual argument shows that V = U ⊕ U^{⊥}, where U^{⊥} = {v ∈ V | b(v, U) = 0} (see, e.g., [34], Chapter 1, Lemma 3.4). Note that in this situation, the nonsingularity of (q, V) implies that of q|_{U^{⊥}}. Using (i) and (ii), the lemma follows immediately by induction.

As a direct consequence, we obtain the following:

5.6 Proposition. There exists an anisotropic nonsingular quadratic space of infinite dimension over F if and only if there exists a chain of anisotropic nonsingular quadratic forms (φ_{n})_{n∈ℕ} over F.

Before we state the analogues of Theorems I and II in characteristic 2, we have to recall a few more definitions and facts. Let W F denote the Witt ring of nonsingular bilinear forms over F, and W_{q}F the Witt group of nonsingular quadratic forms, which is in fact a W F -module. The fundamental ideal of classes of even-dimensional bilinear forms in W F will be denoted by IF, and its n^{th} power by I^{n}F. We put I_{q}^{n}F = I^{n−1}F · W_{q}F. Then I_{q}^{n}F is the submodule of W_{q}F generated (as a group) by the n-fold quadratic Pfister forms

⟨⟨a_{1}, …, a_{n}]] = ⟨1, a_{1}⟩_{b} ⊗ ⋯ ⊗ ⟨1, a_{n−1}⟩_{b} ⊗ [1, a_{n}] ,

with a_{1}, …, a_{n−1} ∈ F^{×} and a_{n} ∈ F; here, we denote a diagonal bilinear form with c_{1}, …, c_{m} in the diagonal by ⟨c_{1}, …, c_{m}⟩_{b}. Quadratic Pfister forms in characteristic 2 have properties quite analogous to those in characteristic different from 2. For example they are either anisotropic or hyperbolic (i.e. isometric to an orthogonal sum of hyperbolic planes). Function fields of nonsingular quadratic forms are defined as in characteristic different from 2, again with the convention that F (ℍ) = F. If q is a nonsingular quadratic form of dimension 2m > 0, then F (q)/F can be realized as a purely transcendental extension of F of transcendence degree 2m − 2 followed by a separable quadratic extension, and F (q)/F is purely transcendental if and only if q is isotropic. Recall that (3.1) and (3.5) remain true in characteristic 2: anisotropic quadratic forms (resp. division algebras of exponent a 2-power) over F stay anisotropic (resp. division) over any odd extension of F and equally over any purely transcendental extension of F. Also, (3.2) stays true in characteristic 2 for nonsingular forms: if π is an anisotropic n-fold quadratic Pfister form and q is any nonsingular form with dim(q) > 2^{n}, then π_{F (q)} is anisotropic. This follows simply by invoking the characteristic 2 analogues of the facts referred to in the proof of (3.2) (see, e.g. [16], Theorem 4.2(i), 4.4). The characteristic 2 version of Theorem I reads as follows.

5.7 Theorem I(2). Let F be a field with char(F) = 2. There exists a field extension K/F with the following properties:

(i) K has no finite extensions of odd degree.

(ii) For any binary nonsingular quadratic form β over K, there is an upper bound on the dimensions of anisotropic nonsingular quadratic forms over K that contain β.

(iii) For any k ∈ ℕ, there is an anisotropic k-fold quadratic Pfister form over K.

In particular, K has infinite u-invariant, I_{q}^{k}K ≠ 0 for all k ∈ ℕ, and any infinite-dimensional nonsingular quadratic space over K is isotropic.

Note that we cannot possibly expect K to be perfect. Indeed, u(F) = ∞ implies û(F) = ∞ and thus [K : K^{2}] = ∞ by (5.4). Using the above mentioned facts on nonsingular forms, quadratic Pfister forms and function fields of nonsingular forms, the proof of Theorem I now easily adapts to become a proof of Theorem I(2). Indeed, it suffices to add the adjective ‘nonsingular’ whenever a quadratic form is mentioned in the proof and to replace ‘Pfister form’ by ‘quadratic Pfister form’ (with the appropriate notation). Also, expressions of type 2^{j} + 1 referring to the dimension of a form must be replaced by 2^{j} +2 as nonsingularity requires even dimension. We leave the details to the reader. To treat the characteristic 2 version of Theorem II, we need a few more facts about quaternion algebras and their products over fields of characteristic 2. A quaternion algebra (a, b]_{F}, with a ∈ F^{×} and b ∈ F, is a 4-dimensional central simple F -algebra generated by two elements x, y subject to the relations x^{2} = a, y^{2} + y = b, xy = (y + 1)x. We now list some relevant facts that allow us to carry over the proofs from characteristic different from 2 to characteristic 2.

5.8 Proposition. Let a_{1} …, a_{n} ∈ F^{×} and b_{1}, …, b_{n} ∈ F be such that A = (a_{1}, b_{1}]_{F} ⊗ ⋯ ⊗ (a_{n}, b_{n}]_{F} is a division algebra. Then the following hold:

(i) The nonsingular (2n + 2)-dimensional quadratic form

φ = [1, b_{1} + ⋯ + b_{n}] ⊥ a_{1}[1, b_{1}] ⊥ ⋯ ⊥ a_{n}[1, b_{n}]

is anisotropic.

(ii) For any field extension K/F of one of the following types, the K-algebra A_{K} = A ⊗_{F} K is a division algebra and φ_{K} is anisotropic:

• K/F is an odd extension;

• K = F (q) where q is a nonsingular quadratic form q such that dim q ≥ 2n + 4 or q ∈ I_{q}^{3}F;

• K/F is purely transcendental.

Proof. (i) This is [23], Proposition 6. (ii) By Part (i) it suffices to prove in each case that A_{K} is a division algebra. For a purely transcendental extension K/F this is obvious, and for an odd extension it is also clear as the index of A is a 2-power. In the case K = F (q), this follows from [23], Theorems 3 and 4.

5.9 Corollary. Suppose that for every n ∈ ℕ there exist a_{1}, …, a_{n} ∈ F^{×} and b_{1}, …, b_{n} ∈ F such that (a_{1}, b_{1}]_{F} ⊗ ⋯ ⊗ (a_{n}, b_{n}]_{F} is a division algebra. Then u(F) = ∞.

The characteristic 2 version of Theorem II reads as follows.

5.10 Theorem II(2). Let F be a field with char(F) = 2. There exists a field extension K/F with the following properties:

(i) K has no finite extensions of odd degree and I_{q}^{3}K = 0.

(ii) For any binary nonsingular quadratic form β over K, there is an upper bound on the dimensions of anisotropic nonsingular quadratic forms over K that contain β.

(iii) For any k ∈ ℕ, there is a central division algebra over K that is decomposable into a tensor product of k quaternion algebras.

In particular, K has infinite u-invariant and every infinite-dimensional nonsingular quadratic space over K is isotropic.

Using (5.8) and (5.9), it is now straightforward to obtain a proof of Theorem II(2) by applying the appropriate changes to the proof of Theorem II, in a similar fashion as was done in the case of Theorem I(2). This time, it is expressions of type 2j + 3 in the proof of Theorem II which must be replaced by 2j + 4 because of the nonsingularity of the forms considered. Again, we leave the details to the reader.

5.11 Remark. In Theorem II (where char(K) ≠ 2), the facts that K has no odd degree extensions and that I^{3}K = 0 but I^{2}K ≠ 0 together imply that K has cohomological dimension cd(K) = 2. In Theorem II(2) (where char(K) = 2) we have again that K has no odd degree extension. This implies in particular that any finite separable extension L/K also has this property, and therefore H^{1}(L, μ_{p}) = L^{×}/L^{×p} vanishes for every finite separable extension L/K and every odd prime p. This implies that cd_{p}(K) = 0 for the cohomological p-dimension of K for any odd prime p (see [37], II.1.2 and II.2.3). On the other hand, cd_{2}(F) ≤ 1 holds for any field F of characteristic 2 (see [37], II.2.2). In our case, there exist anisotropic nonsingular forms of dimension at least 2 over K, thus there certainly are separable quadratic extensions over K. This readily implies that cd_{2}(K) = 1 and therefore cd(K) = sup{cd_{p}(K) | p prime} = 1. However, rather than considering cd_{2}(F) for a field F with char(F) = 2, it is perhaps more meaningful to ask for the separable 2-dimension dim_{2}^{sep}(F) as defined by P. Gille [11]:

dim_{2}^{sep}(F) = sup{r ≥ 0 | H_{2}^{r}(E) ≠ 0 for some finite separable ext. E/F } ,

where the H_{2}^{n}(F) (n ≥ 0) are Kato’s cohomology groups for a field F with char(F) = 2 (see, e.g., [19]). In the situation of Theorem II(2), we have a field K of characteristic 2 with no odd degree extension and I_{q}^{3}K = 0. By Kato’s proof of the Milnor conjecture in characteristic 2 in [19], we have H_{2}^{3}(K) = 0. Furthermore, by Galois theory, if L/K is a finite separable extension then [L : K] is a 2-power and L/K can be obtained as a tower of separable quadratic extensions. But for any field F of characteristic 2 and any separable quadratic extension E/F, we have that H_{2}^{n}(F) = 0 implies H_{2}^{n}(E) = 0 (see, e.g., [4], 6.6). All this together implies that H_{2}^{3}(L) = 0 for every finite separable extension L of K, therefore dim_{2}^{sep}(K) = 2 (note that I_{q}^{2}K ≠ 0).

6 Analogues of the Gross Question

Let (M, ∗, ε) be a monoïd (associative semi-group) with neutral element ε. Let A and S be nonempty subsets of M with ε ∉ S ⊂ A ⊂ M. Denoting by ⟨S⟩ the submonoïd of M generated by S, we furthermore assume that for any a, b ∈ ⟨S⟩, if a ∗ b ∈ A then a, b ∈ A. We now define a U-invariant for this triple (M, A, S) by

U_{M}(A, S) = sup {m ∈ ℕ | ∃s_{1}, …, s_{m} ∈ S with s_{1} ∗ ⋯ ∗ s_{m} ∈ A} .

These definitions have of course been motivated by our investigations of quadratic forms. More precisely, let F be a field with char(F) ≠ 2. Then we take M to be the set of nonsingular quadratic forms (up to isometry) over F, the operation ∗ the orthogonal sum, ε the trivial (0-dimensional) quadratic form, A the set of anisotropic forms over F, and S the set of 1-dimensional (nonzero) quadratic forms over F. In this setting, U_{M}(A, S) is nothing else but u(F). The Gross Question has now an obvious reformulation in this more abstract setting.

6.1 Question. Suppose that U_{M}(A, S) = ∞. Does there exist a sequence (s_{n})_{n∈ℕ} ⊂ S such that s_{1} ∗ ⋯ ∗ s_{n} belongs to A for every n ∈ ℕ?

We proved that this does not always hold for anisotropy of quadratic forms over a field F. We will now pass from quadratic forms to other types of algebraic objects defined over a field that also naturally give rise to a triple (M, A, S), and we will sketch answers to the above question in these new contexts.

Symbol algebras

Let F be a field and n ≥ 2 be an integer. We assume that char(F) does not divide n, and that F contains a primitive n^{th} root of unity ζ which we fix. An F -algebra generated by two elements x, y subject to the relations x^{n} = a, y^{n} = b, xy = ζyx, where a, b ∈ F^{×}, is denoted by (a, b)_{n} and called an nsymbol algebra over F. Note that (a, b)_{n} is a central simple F -algebra of degree n. For n = 2, we recover the case of quaternion algebras. For basic properties of symbol algebras, we refer to [7], §11 (there, such algebras are called ‘power norm residue algebras’). In the sequel, we will concentrate on the case where n = p is a prime number. With F as above, let M be the set of isomorphism classes of central simple algebras over F. The tensor product ⊗, taken over F, endows M with a monoïd structure, where the neutral element is given by the class of F. Let A ⊂ M be the subset of (finite dimensional) central division algebras over F. Further, let S_{p} ⊂ A be the subset given by the non-split p-symbol algebras over F. The Gross Question in this context now becomes the following:

6.2 Question. Suppose that U_{M}(A, S_{p}) = ∞, i.e. suppose that to every n ∈ ℕ there exist p-symbol algebras Q_{1}, …, Q_{n} such that ⨂_{i=1}^{n} Q_{i} is a division algebra. Does there exist a sequence (A_{i})_{i∈ℕ} of p-symbol algebras A_{i} over F such that ⨂_{i=1}^{n} A_{i} is a division algebra for all n ∈ ℕ?

Let us first consider the case p = 2. If we take F = K to be the field constructed in the proof of Theorem II, then we have in fact shown there that U_{M}(A, S_{2}) = ∞, while for any sequence (A_{n})_{n∈ℕ} of quaternion algebras over K, the product A_{1} ⊗⋯⊗A_{n} fails to be a division algebra for n ∈ ℕ sufficiently large. Actually, these two facts do not only follow from the way in which K was constructed, but already from the properties (i)–(iii). We omit the details. Hence, for p = 2, the answer to (6.2) is negative in general. In the sequel, we will sketch how to obtain counterexamples for an arbitrary prime p. Our construction is to some extent similar to the one in the proof of Theorem II, but function fields of quadratic forms will now have to be replaced by function fields of generic partial splitting varieties, also called generalized Severi-Brauer (or Brauer-Severi) varieties, and the special case in (3.4) of Merkurjev’s index reduction results for function fields of quadratic forms will have to be replaced by an appropriate version concerning index reduction for function fields of generic partial splitting varieties. Such generic partial splitting varieties have been studied systematically perhaps for the first time by Heuser [13], and then later by Schofield and Van den Bergh [35], [36], and Blanchet [5]. Blanchet derives in particular an index reduction formula for central simple algebras over function fields of generic partial splitting varieties. This formula has been simplified by Wadsworth [39], and it is that simpler formula which we will use. The reader interested in the most general results on index reduction of central simple algebras over function fields of varieties is referred to the two papers by Merkurjev, Panin and Wadsworth [25], [26].

Let A be a central simple algebra over F of degree n, and let s be a divisor of n. To A we can now associate a generalized Severi-Brauer variety X = SB(A, n, s) such that for any field exension L/F, the L-points X(L) are the sn-dimensional right ideals in A_{L} = A ⊗_{F} L. If L is a splitting field, so that A ⊗_{F} L ≅ End_{L}(V) for an n-dimensional L-vector space V, then X(L) is isomorphic to the Grassmannian Gr(V, s) of s-dimensional subspaces of V. The function field F (X) has the property that ind(A_{F (X)}) divides s, and it is generic for that property in the following sense: If L is any field extension such that ind(A_{L}) divides s, then there exists an F -place F (X) ⟶ L ∪ {∞} (see [13]). More precisely, we have the following (see [5], Proposition 3):

6.3 Lemma. Let A, n, s, X = SB(A, n, s) be as above and let L/F be a field extension. Then the following statements are equivalent:

(i) X has an L-rational point.

(ii) ind(A_{L}) divides s.

(iii) The free compositum L · F (X) is a purely transcendental extension of L.

We now have the following index reduction formula for function fields of generic partial splitting varieties, see [39], Theorem 2:

6.4 Theorem. Let A, n, s, X = SB(A, n, s) be as above, let K = F (X) and let D be a central simple algebra over F. Then

ind(D_{K}) = gcd \{ \frac{s}{gcd(i, s)} ind(D ⊗_{F} A^{−i}) | 1 ≤ i ≤ n \} .

6.5 Corollary. Let p be a prime, let D be a central division algebra of index p^{r} over F, and let A be a central simple algebra of degree p^{m} over F (m ≥ 1) and of exponent dividing p. Let X = SB(A, p^{m}, p^{m−1}). If A is not a division algebra, or if m > r, then D_{F (X)} is a division algebra.

Proof. If A is not a division algebra, then ind(A) divides p^{m−1} and F (X)/F is purely transcendental by (6.3). This clearly implies that D will stay a division algebra over F (X). Now assume that m > r. We apply the above index reduction formula with n = p^{m} and s = p^{m−1}. Let i ∈ {1, …, n}. If p | i, then A^{−i} is split, because exp(A) divides p, and it follows immediately that \frac{s}{gcd(i,s)} ind(D ⊗_{F} A^{−i}) is divisible by ind(D ⊗_{F} A^{−i}) = ind(D). Furthermore, for i = p^{m} we have \frac{s}{gcd(i,s)} ind(D ⊗_{F} A^{−i}) = ind(D). If p |̸ i then gcd(i, s) = 1. Therefore,

\frac{s}{gcd(i, s)} ind(D ⊗_{F} A^{−i}) = p^{m−1} · ind(D ⊗_{F} A^{−i}) ,

and this number is divisible by p^{m−1} and thus by ind(D) = p^{r} ≤ p^{m−1}.

We conclude that

ind(D_{F (X)}) = gcd \{ \frac{s}{gcd(i, s)} ind(D ⊗_{F} A^{−i}) | 1 ≤ i ≤ n \} = ind(D) ,

in other words, D stays a division algebra over F (X).

6.6 Theorem. Let p be a prime and let F be a field with char(F) ≠ p. Then there exists a field extension K/F containing a primitive p^{th} root of unity ζ such that the following holds:

(i) Given a_{0} ∈ K^{×}. Then there exists an ℓ ∈ ℕ depending on a_{0} such that for any a_{1}, …, a_{ℓ}, b_{0}, …, b_{ℓ} ∈ K^{×}, the product ⨂_{i=0}^{ℓ}(a_{i}, b_{i})_{p} is not a division algebra.

(ii) For every n ∈ ℕ there exist p-symbol algebras A_{1}, …, A_{n} over K such that ⨂_{i=1}^{n} A_{i} is a division algebra.

Proof. Let F_{0} = F (ζ) where ζ is a primitive p^{th} root of unity in an algebraic closure of F. Let n ≥ 1 and suppose we have have constructed F_{n−1}. Let now

F_{n−1}^{(n)} = F_{n−1}( X_{1}^{(n)}, Y_{1}^{(n)}, …, X_{n}^{(n)}, Y_{n}^{(n)})

where X_{1}^{(n)}, Y_{1}^{(n)}, …, X_{n}^{(n)}, Y_{n}^{(n)} are indeterminates over F_{n−1}. Let F_{n} denote the free compositum of function fields F_{n−1}^{(n)}(SB(A, p^{j+1}, p^{j})) for all central simple algebras A over F_{n−1} of type A ≅ (a_{0}, b_{0})_{p} ⊗ (a_{1}, b_{1})_{p} ⊗ ⋯ ⊗ (a_{j}, b_{j})_{p} with j < n and a_{0} ∈ F_{j}^{×} and a_{1}, …, a_{j}, b_{0}, …, b_{j} ∈ F_{n−1}^{×}. Finally, we define K = ⋃_{i=0}^{∞} F_{n} and claim that K has the desired properties.

(i) Let a_{0} ∈ K^{×}. Then there exists an integer ℓ > 0 such that a_{0} ∈ F_{ℓ}. Let a_{1}, …, a_{ℓ}, b_{0}, …, b_{ℓ} ∈ K^{×} and consider B = ⨂_{i=0}^{ℓ}(a_{i}, b_{i})_{p}. It suffices to show that B is not a division algebra over K. Now there exists n > ℓ such that a_{1}, …, a_{ℓ}, b_{0}, …, b_{ℓ} ∈ F_{n−1}^{×}, so B is defined over F_{n−1}, and since F_{n−1}^{(n)}(SB(B, p^{ℓ+1}, p^{ℓ})) is part of the compositum F_{n}, we have that ind(B_{F_{n}}) divides p^{ℓ}, which implies that B is not a division algebra over F_{n} and thus also not over K. (ii) For n ≥ 1, consider over F_{n} the algebra

C_{n} = (X_{1}^{(n)}, Y_{1}^{(n)})_{p} ⊗ ⋯ ⊗ (X_{n}^{(n)}, Y_{n}^{(n)})_{p} .

It is well known that C_{n} is a division algebra over F_{n} (see, e.g., [25], Corollary 5.2). Part (ii) now follows if we can show that C_{n} will stay a division algebra over K. This can be achieved by mimicking the argument in part (iii) of the proof of Theorem II, this time by invoking (6.3) and (6.5). We omit the details.

Symbols in Milnor K-theory

Recall the definition of the Milnor K-groups K_{n}F of a field F (see [27]). By definition, K_{0}F = ℤ, and K_{1}F is the multiplicative group F^{×}, written additively with the elements denoted by {a}, a ∈ F^{×}, so that {ab} = {a} + {b} for a, b ∈ F^{×}. For n ≥ 2, K_{n}F is then defined to be the quotient of the tensor product (K_{1}F)^{⊗n} by the subgroup generated by all {a_{1}}⊗⋯⊗{a_{n}} satisfying a_{i} + a_{i+1} = 1 for some i. The image of an element {a_{1}} ⊗ ⋯ ⊗ {a_{n}} in the quotient group K_{n}F is denoted by {a_{1}, …, a_{n}} and called a symbol. We then define the Milnor K-ring as the graded ℤ-algebra K_{∗}F = ⨁_{n=0}^{∞} K_{n}F with multiplication defined on symbols in the obvious way: {a_{1}, …, a_{n}}·{b_{1}, …, b_{m}} = {a_{1}, …, a_{n}, b_{1}, …, b_{m}}. We are interested in K_{n}F/p, the Milnor K-groups modulo p for some prime p. The image of a symbol {a_{1}, …, a_{n}} in K_{n}F/p will again be called a symbol and denoted in the same way. For p = 2, these groups are linked to quadratic form theory through the Milnor Conjecture (now a theorem due to Orlov, Vishik, and Voevodsky [29]) which asserts that if char(F) ≠ 2 then K_{n}F/2 is isomorphic to I^{n}F/I^{n+1}F, via an isomorphism that maps {a_{1}, …, a_{n}} to the class of ⟨⟨a_{1}, …, a_{n}⟩⟩ modulo I^{n+1}F. We now consider the abstract version of the Gross Question (6.1) in the following setting, where we assume F^{×} ≠ F^{×p} because otherwise K_{n}F/p = 0 for all n ≥ 1. Let M = K_{∗}F/p, S = {{a} | a ∈ F^{×} \ F^{×p}} (this is nonempty by assumption), A = {{a_{1}, …, a_{n}} ≠ 0 | n ∈ ℕ, a_{i} ∈ F^{×}}. It is obvious that for n ≥ 1 we have K_{n}F/p ≠ 0 if and only if there exist a_{1}, …, a_{n} ∈ F^{×} with {a_{1}, …, a_{n}} ≠ 0. In this setting, Question (6.1) becomes:

6.7 Question. Suppose that U_{M}(A, S) = ∞, i.e. K_{n}F/p ≠ 0 for all n ∈ ℕ. Does there exist a sequence (a_{n})_{n∈ℕ} ⊂ F^{×} such that {a_{1}, …, a_{n}} ≠ 0 for every n ∈ ℕ?

Let us first consider the case where char(F) = p. Then the answer to the above question is positive by the following:

6.8 Proposition. Let F be a field of characteristic p > 0. Then the following are equivalent:

(i) [F : F^{p}] = ∞.

(ii) K_{n}F/p ≠ 0 for all n ∈ ℕ.

(iii) There exists a sequence (a_{n})_{n∈ℕ} ⊂ F^{×} such that {a_{1}, …, a_{n}} ≠ 0 for every n ∈ ℕ.

For p = 2, the above statements are further equivalent to any of the following:

(iv) û(F) = ∞.

(v) sup {dim(b) | b anisotropic symmetric bilinear form over F } = ∞.

(vi) There exists an infinite-dimensional anisotropic quadratic space over F.

(vii) There exists an infinite-dimensional anisotropic symmetric bilinear space over F.

Proof. Recall that a subset T ⊂ F is called p-independent if, for any finite subset {a_{1}, …, a_{n}} ⊂ T, one has [F^{p}(a_{1}, …, a_{n}) : F^{p}] = p^{n}, and that T ⊂ F is called a p-basis of F if T is a minimal generating set of the extension F/F^{p}, i.e. F = F^{p}(T) and T is p-independent. The key observation here is the fact that for a_{1}, …, a_{n} ∈ F^{×} we have that {a_{1}, …, a_{n}} ≠ 0 if and only if a_{1}, …, a_{n} are p-independent, in other words [F^{p}(a_{1}, …, a_{n}) : F^{p}] = p^{n}. This is an immediate consequence of the Bloch-Kato-Gabber Theorem (see [6], Theorem 2.1, or [17], Appendix A2). The equivalence of the first three statements is now immediate and we leave the details to the reader. For p = 2 it readily follows from (5.3) and (5.4) that (i) is equivalent to any of the statements (iv) to (vii).

Let us now turn to the case char(F) ≠ p. For p = 2, the answer to the above question will be negative in general, i.e. there are fields such that K_{n}F/2 ≠ 0 for all n ∈ ℕ, but for any sequence (a_{n})_{n∈ℕ} ⊂ F^{×} one has {a_{1}, ⋯, a_{m}} = 0 for sufficiently large m. Indeed, any field as constructed in Theorem I will do. To see this, it suffices to note that the map from the set of isometry classes of n-fold Pfister forms over F into K_{n}F/2 given by ⟨⟨a_{1}, …, a_{n}⟩⟩ ↦ {a_{1}, …, a_{n}} is well-defined, injective, and sends the hyperbolic Pfister form to zero, see [8], Main Theorem 3.2 (here, we do not need the full thrust of the Milnor Conjecture). We leave the details to the reader. Now if p ≠ 2 (and char(F) ≠ p), we believe (but have not checked) that in general the answer to the above question should be negative as well. To construct counterexamples, it seems reasonable to try a similar approach as in our other constructions using a tower of iterated function fields. Candidates for these functions fields will naturally be function fields of (generic) splitting varieties of symbols in Milnor K-theory modulo p. The norm varieties as constructed by Rost (see [33], also [18]) provide examples for such splitting varieties.

Acknowledgements

We wish to express our gratitude to Pasquale Mammone for inspiring discussions on the subject of this work. He brought to our attention the intimate connection of the problem as we have originally formulated it with the Gross Question. We further wish to acknowledge financial support provided by the European RTN Network ‘Algebraic K-Theory, Linear Algebraic Groups and Related Structures’ (HPRN CT-2002-00287). We also thank the referee for some helpful comments.

References

[1] J.K. Arason, Cohomologische Invarianten Quadratischer Formen. J. Algebra 36 (1975), 448–491.

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

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

[4] R. Aravire, R. Baeza, Milnor’s K-theory and quadratic forms over fields of characteristic two. Comm. Algebra 20 (1992), 1087–1107.

[5] A. Blanchet, Function fields of generalized Brauer-Severi varieties. Comm. Algebra 19 (1991), 97–118.

[6] S. Bloch, K. Kato, p-adic étale cohomology. Inst. Hautes Études Sci. Publ. Math. 63 (1986), 107–152.

[7] P.K. Draxl, Skew Fields. London Mathematical Society Lecture Notes series, vol. 81, Cambridge University Press, Cambridge 1983.

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

[9] R. Elman, T.Y. Lam, Quadratic forms and the u-invariant I. Math. Z. 131 (1973), 283–304.

[10] R. Elman, T.Y. Lam, A.R. Wadsworth, Orderings under field extensions. J. reine angew. Math. 306 (1970), 7–27.

[11] P. Gille, Invariants cohomologiques de Rost en caractéristique positive. K-Theory 21 (2000), 57–100.

[12] H. Gross, Quadratic forms in infinite-dimensional vector spaces. Birkhäuser, Boston/Basel/Stuttgart, 1979.

[13] A. Heuser, Partielle Zerfällung einfacher Algebren. PhD thesis, Universität Regensburg, 1976.

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

[15] D.W. Hoffmann, Dimensions of Anisotropic Indefinite Quadratic Forms, I. Documenta Math., Extra Volume, Quadratic Forms LSU (2001), 183–200.

[16] D.W. Hoffmann, A. Laghribi, Quadratic forms and Pfister neighbors in characteristic 2. Transactions Amer. Math. Soc. 356 (2004), 4019–4053.

[17] O. Izhboldin, p-primary part of the Milnor K-groups and Galois cohomologies of fields of characteristic p. With an appendix by Masato Kurihara, Ivan Fesenko. Invitation to higher local fields (Münster, 1999), 19–41. Geom. Topol. Monogr. 3, Geom. Topol. Publ., Coventry, 2000.

[18] S. Joukhovitski, A. Suslin, Norm varieties. J. Pure Appl. Algebra 206 (2006), 245–276.

[19] K. Kato, Symmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic two. Invent. math. 66 (1982), 493–510.

[20] T.Y. Lam, The algebraic theory of quadratic forms. W. A. Benjamin, Inc., Reading, MA, 1973.

[21] T.Y. Lam, Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67, Amer. Math. Soc., Providence, RI, 2005.

[22] P. Mammone, R. Moresi, A.R. Wadsworth, u-invariants of fields of characteristic 2. Math. Z. 208 (1991), 335–347.

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

[24] A.S. Merkurjev, Simple algebras and quadratic forms. Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 218–224; English translation in: Math. USSR-Izv. 38 (1992), 215–221

[25] A.S. Merkurjev, I.A. Panin, A.R. Wadsworth, Index reduction formulas for twisted flag varieties. I. K-theory 10 (1996), 517–596.

[26] A.S. Merkurjev, I.A. Panin, A.R. Wadsworth, Index reduction formulas for twisted flag varieties. II. K-theory 14 (1998), 101–196.

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

[28] J. Mináč, A. Wadsworth, The u-invariant for algebraic extensions, Ktheory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), AMS, Providence, RI, 1995, 333–358.

[29] D. Orlov, A. Vishik, V. Voevodsky, An exact sequence for K_{∗}^{M}/2 with applications to quadratic forms. Ann. of Math. 165 (2007), 1–13.

[30] A. Pfister, Quadratic forms with applications to algebraic geometry and topology. London Mathematical Society Lecture Notes series, vol. 217, Cambridge University Press, Cambridge, 1995.

[31] R.S. Pierce, Associative algebras. Graduate Texts in Mathematics 88, Springer-Verlag, New York, 1982.

[32] A. Prestel, Lectures on formally real fields. Lecture Notes in Mathematics 1093, Springer-Verlag, Berlin, 1984.

[33] M. Rost, Norm varieties and algebraic cobordism. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 77–85, Higher Ed. Press, Beijing, 2002. (Errata: Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 649, Higher Ed. Press, Beijing, 2002.)

[34] W. Scharlau, Quadratic and Hermitian forms. Grundlehren 270, Springer-Verlag, Berlin, 1985.

[35] A. Schofield, M. Van den Berg, The index of a Brauer class on a Brauer-Severi variety. Trans. Amer. Math. Soc. 333 (1992), 729–739.

[36] A. Schofield, M. Van den Berg, Division algebra coproducts of index n. Trans. Amer. Math. Soc. 341 (1994), 505–517.

[37] J.-P. Serre, Galois Cohomology. Springer-Verlag, Berlin Heidelberg, 1997.

[38] J.-P. Tignol, Réduction de l’indice d’une algèbre simple centrale sur le corps des fonctions d’une quadrique. Bull. Soc. Math. Belg. Sér. A 42 (1990), 735–745.

[39] A.R. Wadsworth, The index reduction formula for generic partial splitting varieties. Comm. Algebra 21 (1993), 1063–1070.

Karim Johannes Becher Fachbereich Mathematik und Statistik, D203 Universität Konstanz D-78457 Konstanz Germany karim.johannes.becher@unikonstanz.de Detlev W. Hoffmann School of Mathematical Sciences University of Nottingham University Park Nottingham NG7 2RD UK detlev.hoffmann@nottingham.ac.uk