Abstract

For a (formally) real field , the vanishing of a certain power of the fundamental ideal in the Witt ring of implies that the same power of the fundamental ideal in the Witt ring of is torsion free. The proof of this statement involves a fact on the structure of the torsion part of powers of the fundamental ideal in the Witt ring of . This fact is very difficult to prove in general, but has an elementary proof under an assumption on the stability index of . We present an exposition of these results.

1. Introduction

In the study of sums of squares and quadratic forms over fields, it has often been fruitful to consider the quadratic extension obtained by adjoining . Consider, for example, an extension of transcendence degree of a real closed field . Tsen-Lang theory yields that quadratic forms over of dimension greater than are isotropic. From this, one can derive that in any totally positive element is a sum of squares and that torsion quadratic forms over of dimension greater than are isotropic. The proof of the statement on sums of squares in uses the observation that a certain field property goes down a quadratic extension.

We assume that the reader is familiar with the classical theory of quadratic forms over fields as presented in [1]. We use standard notation and terminology. When we speak about quadratic forms we always assume them to be regular, and we often just call them forms.

Let be a field of characteristic different from 2. We denote by the multiplicative group of and by the subgroup of the nonzero sums of squares in . By the Artin-Schreier Theorem, admits a field ordering if and only if ; in this case, we say that is real, otherwise nonreal.

Let denote the Witt ring of and its fundamental ideal. Let be the torsion part of . If is real, then , otherwise .

Let be a positive integer. We put and . Following [2] and [1, Chapter XI, Section 4], we denote by the condition that every torsion -fold Pfister form over a given field is hyperbolic. Recall that is generated (as a group) by the -fold Pfister forms over . Hence, if is a nonreal field, then satisfies if and only if . It was conjectured in [2] that, without assuming that is nonreal, the following holds.

Conjecture 1.1. The field satisfies if and only if .

This is obviously implied by the following stronger conjecture, which was proved in [3, (2.8)] as a consequence of Voevodsky's proof of the Milnor Conjecture and the deep results in [4].

Conjecture 1.2. One has .

Nevertheless, one may wish to have more elementary proofs of these conjectures, even for special cases. For , such a proof for Conjecture 1.1 can be found in [1, Chapter XI, (4.1) and Chapter XII, (3.1)]. For an elementary proof of Conjecture 1.2 for , see [1, Chapter XI, (4.2)] or Proposition 3.2 below. For arbitrary , Krüskemper showed in [5, Theorem 3] by elementary arguments that a sum of up to three scaled -fold Pfister forms lies in if and only if it lies in ; in Theorem 3.8, we give a self-contained exposition of this result.

There is another situation where an elementary proof of Conjecture 1.2 is possible, depending on the stability index of the field . Following [6], the (reduced) stability index of is defined as Note that if and only if is either nonreal or uniquely ordered.

In [7], Krüskemper outlined an elementary proof of Conjecture 1.2 for the case where , using several facts stated in [8] without a detailed proof.

Our efforts to understand the mentioned results from [5, 7] led us to a more detailed and structured exposition, which we want to make available to a wider audience, though we do not claim any originality of the ideas. Shortly after completing this note, we learned that another exposition of Krüskemper's proof that Conjectures 1.2 and 1.1 hold for appeared in [9, Section 35.B]. Here, following [7], we work within the more general framework of preorderings and the reduced theory of quadratic forms.

Let us point out that having Conjecture 1.2 for leads to a proof of the following statement, as observed in [5, Proposition 1].

Theorem 1.3 (Elman-Lam-Krüskemper). For , the following statements are equivalent:
(i).
(ii) and .
(iii) and .

Assuming that , it is clear that holds if and only if . Therefore, (ii) and (iii) are equivalent. Elman and Lam showed in [2, (4.4), (4.5), (4.7)] that if and only if satisfies and (see also [1, Chapter XI, (4.7), (4.14), (4.18)]); this fact was rediscovered by Elman and Prestel in [10, (3.3)]. Hence, to complete the proof of Theorem 1.3, it is sufficient to know that Conjecture 1.1 holds when .

Finally, note that the equivalent conditions in Theorem 1.3 hold in the case where is an extension of a real closed field of transcendence degree strictly smaller than . In fact, since any quadratic form over of dimension exceeding is isotropic, it is clear that , and [2, (6.1)] gives a more direct way to show that in this case.

2. Preorderings

We recall some facts from real algebra and the reduced theory of quadratic forms modulo a preordering. For details, we refer to [11]. Let denote the set of all field orderings of . For a quadratic form over and , we denote by the signature of at .

A preordering of is a proper subset that contains the squares in and that is closed under addition and multiplication in . Note that these conditions entail that and , in particular is real. Conversely, if is real, then has preorderings and the smallest one is .

Let be a fixed preordering of . We denote . We denote by the set of field orderings of that contain . By [1, Chapter VIII, (9.6)], we have . The set is considered as a topological space with the Harrison topology, which is generated by the sets where . For and , we put the sets of this shape are called Harrison sets, and they form a base of the Harrison topology on . For a nontrivial Harrison set of , the smallest such that with is called the degree of ; the trivial Harrison sets and are given the degree 0.

We write and denote the kernel of this homomorphism by . (Note that in [11] this notation is used for what is in our notation: the fundamental ideal in the reduced Witt ring .) For , let .

Let be a quadratic form over . We say that is -positive if . We say that is -isotropic if is isotropic for some -positive form over ; by [11, (1.20)], this is equivalent to having for some form over with . We write for the set of elements such that for some -positive form over . By [11, (1.16)], if and only if there is a -positive Pfister form over such that is hyperbolic. Moreover, the ideal is generated by the binary forms with , by [11, (1.26)]. A form over with and is said to be -isometric to ; by [11, (1.19)] this implies that . These facts will be used in the next section.

The stability index of is defined as This is reconcilable with the above definition of the stability index of a field. In fact, if is real then for the preordering , we have by [11, (13.1)], and furthermore and for .

3. Krüskemper's Theorem

Let be a preordering on and . A more general version of Conjecture 1.2 is the following conjecture.

Conjecture 3.1. One has .

This conjecture is proved in [3, (2.7)], but the proof of the nontrivial inclusion uses the deep results in [4]. In [7], Krüskemper sketches an elementary proof of Conjecture 3.1 for . We want to provide a complete exposition of Krüskemper's result, which we achieve with Theorem 3.10. On the way, we shall obtain with Theorem 3.8 that, for any , a sum of up to three scaled -fold Pfister forms over belongs to if and only if it belongs to .

We begin with the elementary proof of Conjecture 3.1 for , adapted from [1, Chapter XI, (4.1)].

Proposition 3.2. One has .

Proof. For and , one has . For , one has . As the ideal is generated by the forms with , any element of is given by a form , where , , and with . Using induction on and the above congruences shows that .

Lemma 3.3. Let and be Pfister forms over , and let be the form over such that . Then, for , there exists a Pfister form over with sign_Tsign_T.

Proof. There exists a -positive Pfister form such that . By [1, Chapter X, (1.10)], we have for a Pfister form , and then holds.

The next two statements are contained without proof in [8, Section 5].

Lemma 3.4 (Marshall). Let be an -fold Pfister form over where . If , then there is an -fold Pfister form over such that . If , then .

Proof. We use induction on . For , the first part is trivial. In the induction step, we will use both parts of the statement for to prove the first part for . Before we do so, we explain how, for given , the second part of the statement follows from the first part. This together will give the proof.
Let . Then, for some . By Proposition 3.2, we have . If now for an -fold Pfister form , then and thus .
For the induction step, assume now that and write with and an -fold Pfister form . Let . Since , there exist and such that and thus . By the induction hypothesis, we then have for some -fold Pfister form , and . We obtain and then choose .

Lemma 3.5 (Marshall). Assume that , and let and be two -fold Pfister forms over . If sign_Tsign_T, then .

Proof. The proof is by induction on . The claim is trivial for . Write with and an -fold Pfister form . As and are -isometric, we have , and thus Lemma 3.4 yields that for some -fold Pfister form . If , then for . Assume now that . Then is a preordering. For any , there are with , and then This shows that . Since , we have by the induction hypothesis. Multiplying with , we now obtain that .

Corollary 3.6. Let be an -fold Pfister form over , and a form over of even dimension. Then, if and only if .

Proof. One implication is clear as trivially . Assume that . Then, there exists a -positive Pfister form such that is hyperbolic. For , the -fold Pfister form lies in the annihilator of , thus , and by Lemma 3.5 then . By [1, Chapter XI, (3.1)], the binary forms with generate the subideal of annihilated by , and since belongs to this ideal, we conclude that .

Lemma 3.7. Assume that , and let be -fold Pfister forms over and such that sign_Tsign_Tsign_T. Then, there exists an -fold Pfister form over and with such that for .

Proof. Choose an -fold Pfister form and -fold Pfister forms such that for , where and where is as large as possible in this range.
We claim that . Suppose on the contrary that . Then, the form is -isotropic, given that and . So, there exists an element , and Lemma 3.3 now gives a contradiction to the maximality of . Hence, and are 1-fold Pfister forms.
We choose such that for . Then, we have for . Let . It follows that Since the signatures of the Pfister forms and lie in , we thus have for . Now, Lemma 3.5 yields that for .

The following result appears in [5, Theorem 3].

Theorem 3.8 (Krüskemper). Let . Let for -fold Pfister forms over and . Then, sign_T if and only if .

Proof. If , then obviously . Assume now that . Then, , and comparing dimensions we conclude that is -isotropic, so there is an element . Using Lemma 3.4, we obtain that for . Hence, with , we have By Lemma 3.7 applied with , there is an -fold Pfister form over and with such that for . It follows that As , we obtain that . By Lemma 3.5, then , so .

Given a Pfister form over , we put . Hence, for and , we have with .

Proposition 3.9. Assume that . Every class of is given by a form where , are forms over , and are -fold Pfister forms over such that are nonempty and pairwise disjoint.

Proof. Consider and an -fold Pfister form over . As , there exist -fold Pfister forms and over such that It follows that . Applying Theorem 3.8 with and yields that
Consider now a finite set . For , let denote the Harrison set consisting of the orderings with and . Let , the power set of . Then, is a finite partition of . (However, can be empty for many .) Induction on the cardinality of , using the above argument in the induction step, yields that, given an arbitrary -fold Pfister form over , there exists a family of -fold Pfister forms such that for every and Moreover, if for and , then in case all belong to and otherwise.
An arbitrary element of is given by a form where , and where are -fold Pfister forms over . Choose a finite set such that for there exist such that . Put . Using that , we choose a family of -fold Pfister forms such that for .
By the above argument, we choose for a family of -fold Pfister forms such that and such that is either empty or equal to for any . Using Lemma 3.5, for and , it follows that if and otherwise. Put . For , let be the orthogonal sum of those with such that . We conclude that

Theorem 3.10 (Krüskemper). If , then .

Proof. We trivially have that . To prove the converse inclusion, we assume that and . By Proposition 3.9, we have where , are forms and where are -fold Pfister forms such that are nonempty and pairwise disjoint. Using this and the fact that , we obtain for that and that because , so that by Corollary 3.6. Therefore, .

Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (Project Quadratic Forms and Invariants, no. BE 2614/3-1) and by the Zukunftskolleg, Universität Konstanz.