Isotropy of Quadratic Spaces in Finite and Infinite Dimension

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 exam- ples of such fields and also discuss related problems in the theory of central simple algebras and in Milnor K-theory.


Introduction
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 of characteristic different from 2. For a nonreal1 field F it 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.
A quadratic space over the 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) = λ2 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. 2   If V has finite dimension n, we 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.By this identification, we will also call (V, q) a quadratic form over F if dim V < ∞.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.Assume now that the quadratic space (V, q) over F is anisotropic.For any positive integer n ≤ dim(V ), we may choose an n-dimensional subspace V n of V and consider the restriction q n of q to V n ; in this way we obtain an n-dimensional quadratic form (V n , q n ) which is obviously again anisotropic.By this simple argument, we see 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 in the beginning, 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] (that appeared 1979) by the following sample of 'a number of pretty and unsolved problems' in this area, which we state in his terms (cf.[12], p. 3; here, it is assumed that the characteristic is different from 2): 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, i.e. a field where −1 is a sum of squares; for over a real field anisotropic quadratic spaces of infinite dimension do always exist.Indeed, one observes (or one may take it as a definition) that the field F is real if and only if the infinite-dimensional quadratic space (V, q) given by V = k (N) and q : V −→ k, (x i ) −→ x 2 i is anisotropic.
By restricting to those quadratic spaces that are totally indefinite (i.e.indefinite with respect to every field ordering), we obtain a meaningful analogue of the Gross Question, to which we can equally provide a solution.
Of course, one may place the Gross Question also in the context of characteristic 2; there, however, one has to distinguish between bilinear forms and quadratic forms.When considering quadratic forms, one furthermore has to distinguish the case of nonsingular forms from the case where one allows 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 characteristic 2 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 of characteristic different from 2 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 kth 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 funcion 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 analoguous 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 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( √ −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 If ϕ is a quadratic form over F and n ∈ N, we denote by n × ϕ the n-fold orthogonal sum ϕ ⊥ • • • ⊥ ϕ.Given two quadratic forms ϕ and ψ over F , we write ψ ⊂ ϕ to indicate that ψ is a subform of ϕ, in other words, that there exists another quadratic form τ over F such that ϕ ∼ = ψ ⊥ τ .A quadratic space (V, q) is said to be nonsingular if the radical 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.
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 finitedimensional.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.

The u-invariant and its relatives
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 finite-dimensional 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 .
From these considerations we conclude: 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 n × ϕ is hyperbolic for some n ≥ 1.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 } .
Here, 'torsion' means that the Witt class of ϕ is a torsion element in the Witt ring W F .It is well known that if F is nonreal, then any form over F is torsion, hence the above supremum is actually taken over all anisotropic forms over F in this case.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} .
2.2 Proposition.If there exists an infinite-dimensional quadratic space over F , then u ′ (F ) = ∞.
Proof: Assume that there exists an infinite-dimensional quadratic space over F which is anisotropic.Then there is also a chain (ϕ n ) n∈N of anisotropic forms over F .Now we have certainly u(ϕ n , F ) = ∞ for any n ∈ N, and this implies that 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.
By the last two statements, any nonreal field F such that 0 < u ′ (F ) < ∞ will provide an example which 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 ) .
Proof: The straightforward proof is based on the well known relationship between the quadratic forms over F and over F ((t)) (see [20], Chapter VI, Proposition 1.9).The details are left to the diligent reader.
Using this lemma together with the theorems mentioned in the introduction, one has the following result.
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 or Theorem II, respectively, asserts the existence of such fields for m = 0.The induction step from m to m + 1 is clear from the above 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 Throughout this section, all fields are assumed to be of characteristic different from 2.
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. One can immediately generalise this to 'odd' algebraic extensions that are not necessarily finite.
For the following definition we allow the case where F has characteristic 2, for later reference.Note that Springer's Theorem is independent of the characteristic of F .
An algebraic extension L/F is called an odd closure of 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.
Using Springer's Theorem, we readily obtain: 3.1 Lemma.Let K/F be an odd extension.Then any anisotropic form over F stays anisotropic over 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)).Therefore, we obtain in the same way as above: 3.2 Lemma.Let K/F be an odd extension.Then any central division algebra of exponent 2 over F remains a division algebra after scalar extension to K.
We now turn to extensions of the type F (ϕ)/F , where F (ϕ) is the function field of a quadratic form ϕ over F .
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).
3.4 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 (3.3).
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. (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

Theorem
3.6 Remark.Statements analoguous to (3.1) and (3.2) hold for purely transcendental extensions.More precisely, if the field extension K/F is purely transcendental, then every anisotropic quadratic form over F stays anisotropic over K and every division algebra over F extends to a division algebra over K.We will use this fact repeatedly, especially in the case where K is the function field of an isotropic quadratic form over F .Indeed, if a quadratic form ϕ over F is isotropic, then F (ϕ)/F is purely transcendental of transcendence degree dim(ϕ) − 2 (see [34], 4.5.2(vi)).
We are now ready for the proofs of the first two theorems formulated in the introduction.

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 where n are indeterminates over F # n−1 .We define F n as the free compositum 3 of all the function fields F (n) n−1 (ϕ) where ϕ is an anisotropic form defined over F n−1 such that, for some j < n, ϕ contains a binary form defined over F j and dim(ϕ) = 2 j + 1.
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) n−1 (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 n−1 , respectively, with the algebraic closure of F j .
Let us assume from now on that n > j.Note that F , 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 as a free compositum of certain function fields F (n) n−1,j (ϕ) where ϕ is a form defined over F (n) n−1,j which is either of dimension at least 2 j+1 +1 or which contains a binary form defined over F j and thus is isotropic over F Consider now an anisotropic m-fold Pfister form π defined over F (n) n−1,j , where m ≤ j + 1.Using (3.3) and (3.6) it follows, that π stays anisotropic over F n,j , again by (3.6).But then π stays anisotropic over F (n+1) n,j as well.Repeating this, we see that π stays anisotropic over all the fields F n,j when j is fixed and n increases.
Let now k be any positive integer.Let π denote the k-fold Pfister form we know that π is still anisotropic when considered as a form over the field F n ).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.

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 where • dim(ϕ) = 2j + 3 for some j < n 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 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) excludes the possibility that there is an infinite-dimensional anisotropic quadratic space over K.As before, the field K is perfect by (i) and nonreal by (iii).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 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 n−1 , 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 ), a purely transcendental extension of F # n−1,j , which in turn is an odd extension of F n−1,j .Using (3.2) and (3.6), it follows that every division algebra of exponent 2 over F n−1,j remains a division algebra after scalar extension to F where ϕ is a form defined over F (n) n−1,j 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 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) j−1 for some j ∈ N. Assume that for some n > j, the algebra D will stay a division algebra after extending scalars to F (n) n−1,j .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+1) n,j . Repeating this argument shows that D will stay a division algebra after scalar extension to F (N ) N −1,j for any N ≥ n.
Let now k be a positive integer and let D denote the tensor product of quaternion algebras (X are algebraically independent over the field F k−1 , hence also over its algebraic closure But then D Fn 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.
3.9 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 we end up with is countable and thus can be embedded into C.

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.
We consider an ordering P on F and denote by < P the corresponding order relation on F .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.
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].
In view of Theorem IV, we 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 The following observation is useful when dealing with infinite-dimensional totally indefinite quadratic spaces.
4.1 Proposition.Every totally indefinite quadratic space over 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 [32], Theorem 6.5).We put a P = q(v P ) for every P ∈ X F .The above choice of the family (v P ) P ∈X F ⊂ F implies that 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 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.

Proof of Theorem III.
This time, 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.Let F 0 = F and let P 0 be any ordering of this field.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 where n−1 be any ordering on F • 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) n−1 and by the construction of our orderings, all the above forms are in fact totally indefinite at P (n) n−1 .By (4.2), the ordering P (n) n−1 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.
(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) imply that the cohomological dimension of K( √ −1) is at most 2. That it is exactly 2 then follows from (iv).
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 the direct limit of the orderings 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.

Proof of Theorem IV.
Again, 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.
Let F 0 be the given real field F and P 0 any ordering of this field.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) n−1 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 ) at which X (n) is positive.We define F n to be the free compositum of all function fields F (n) n−1 (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) n−1 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 (in fact 2 n ) squares in K.This shows that K is uniquely ordered.To show that P is this unique ordering, it therefore suffices to show that P consists exactly of all nonzero sums of squares.
Any sum of squares s ∈ K × is already a sum of squares in F n for some n and hence in P n .Thus, s ∈ P .Conversely, any s ∈ P is in P n for some n, which by the above reasoning implies that s is a sum of (in fact 2 n ) squares in K.
(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 n−1 , 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 where ϕ is a form defined over F (n) n−1,j which either is 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 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).

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 concerning quadratic and bilinear forms in characteristic 2.
Let (V, q) be a quadratic space over a field F of characteristic 2, and let b q : V × V → F be the 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 The quadratic space (V, q) is said to be 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 s i=1 a i X 2 i 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 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 u, respectively.They are defined as follows: u(F ) = sup{dim(q) | q anisotropic nonsingular quadratic form over F } u(F ) = sup{dim(q) | q anisotropic quadratic form over F } Clearly, we have u(F ) ≤ u(F ), and u(F ) is always even if finite.
One can 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 just coincide with [F : F 2 ], the degree of inseparability of F .
We will now concentrate for a moment on totally singular quadratic spaces.These are, in fact, very easy to treat.
For a field F of characteristic 2 we fix an algebraic closure F and put . Hence the squaring map sq : x → x 2 yields a quadratic map sq F : √ F → F over F , and the quadratic space ( 5.1 Proposition.Let F be a field of characteristic 2. The quadratic space ( √ F , sq F ) is anisotropic and totally singular.Any anisotropic totally singular quadratic space over F is isometric to a subspace of ( √ 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 Since q is totally singular, ρ is a homomorphism of F -vector spaces 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 ( √ 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.

Proof:
If [F : F 2 ] < ∞, then the claim is obvious from the previous results in this section.If [F : F 2 ] = ∞ then the same results yield the existence of infinite-dimensional F -spaces (e.g.√ F ) carrying anisotropic totally singular quadratic forms and anisotropic symmetric bilinear forms.By restricting to subspaces of dimension n for arbitrary n ∈ N, we obtain that the corresponding suprema for the anisotropic dimensions of totally singular forms and symmetric bilinear forms are infinite.
We next consider general quadratic forms in characteristic 2 and the corresponding u-invariant.The first part of the following statement is [22], Corollary 1.

Proposition.
Let F be a field of characteristic 2. Then Furthermore, u(F ) = ∞ if and only if there exists a totally singular anisotropic quadratic space of infinite dimension over F .Proof: The first inequality is obvious from the last corollary.To prove the second inequality, we may assume that [F : F 2 ] < ∞.Let q be an anisotropic quadratic form over F .We may write Since this form is anisotropic, the totally singular subform a 1 , . . ., a r , c 1 , . . ., c s is anisotropic as well, whence r + s ≤ [F : F 2 ] and thus dim(q) ≤ 2 [F : . The last part of the statement now also follows from the last corollary.
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 contains a subspace isometric to the direct limit 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) Suppose U ⊂ V is a 2m-dimensional subspace with ϕ = q| U nonsingular.Let V = U ⊕W and let (w i ) i∈I be a basis of W .Let (x 1 , y 1 , . . ., x m , y m ) be a basis of U such that, with respect to this basis, ϕ For each i ∈ I, let we thus have q ∼ = ϕ ⊥ ϕ ⊥ with ϕ nonsingular of dimension 2m and ϕ ⊥ nonsingular.Using (i) and (ii), the lemma follows immediately by induction.
As a direct consequence, we obtain the following:

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 Fmodule.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 n q F = I n−1 F • W q F .Then I n q F is the submodule of W q F generated (as a group) by the n-fold quadratic Pfister forms 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.Lemma (3.1) is still true in characteristic 2, i.e. an anisotropic quadratic form (possibly singular) stays anisotropic over any odd extension of F and equally over any purely transcendental extension of F .Also, (3.3) stays true in characteristic 2 for nonsingular forms.More precisely, if π is an anisotropic n-fold quadratic Pfister form and q is any nonsingular form of dimension > 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 (3.3), or in (3.4).See, e.g.[16], Theorem 4.2(i), 4.4, for the precise formulation in characteristic 2 of these facts.
We can now state the characteristic 2 version of Theorem I.

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 k q 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 u(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 simply 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 tensor 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.
(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 3 q F ; • K/F is purely transcendental.
(ii) By Part (i) it suffices to prove 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; for 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 The characteristic 2 version of Theorem II now 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 3 q 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,II.2.3).
On the other hand, cd 2 (F ) ≤ 1 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 sep 2 (F ) as defined by P. Gille [11]: where the H n 2 (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 3 q K = 0.By Kato's proof of the Milnor conjecture in characteristic 2 in [19], we have H 3 2 (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 n 2 (F ) = 0 implies H n 2 (E) = 0 (see, e.g., [4], 6.6).All this together implies that H 3 2 (L) = 0 for every finite separable extension L of K, therefore dim sep 2 (K) = 2 (note that I 2 q K = 0).

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 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.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 n-symbol 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 such 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 n i=1 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 n i=1 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.
So for p = 2, the answer to (6.2) is negative in general.We will sketch in the sequel that there are counterexamples for arbitrary primes 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 (3.5) 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 the latter 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. In the case where 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.
(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 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 We conclude that ind(D F (X) ) = gcd 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) For any a 1 ∈ K × , there exists an n ∈ N such that for any choice of a 2 , . . ., a n , b 1 , . . .b n ∈ K × , the product n i=1 (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 n i=1 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 n , Y (i) Let a 1 ∈ K × .Then there exists j ∈ N such that a = a 0 ∈ F j .Let a 1 , . . ., a j , b 0 , . . ., b j ∈ K × and consider B = j i=0 (a i , b i ) p .It suffices to show that B is not a division algebra over K. Now there exists n > j such that a 1 , . . ., a j , b 0 , . . .b j ∈ F n−1 , so B is defined over F n−1 , and since F (n) n−1 (SB(B, p j+1 , p j )) is part of the compositum F n , we have that ind(B Fn ) divides p j , 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 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.
(n) n−1,j .Hence, by Merkurjev's Criterion (3.5) and by (3.6), any division algebra over F (n) extending P # n−1 .Let now F n be the free compositum of the function fields F (n) n−1 (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 (n)n−1,j .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) n−1,j 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 overF (k) k−1 and is still anisotropic over F (k) k−1,k−1 .It follows now from the above arguments that this form stays anisotropic over F n,k−1 , for all n > k.

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?

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 s gcd(i,s) ind(D ⊗ F A −i ) = ind D. If p | i then gcd(i, s) = 1.Therefore, 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 .
indeterminates over F n−1 .Let F n denote the free compositum of function fields F (n) n−1 (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 n , b 0 , . . ., b n ∈ F (n)× n−1 .Finally, we define K = ∞ i=0 F n and claim that K has the desired properties.