Dimensions of Anisotropic

Indefinite Quadratic Forms, I

Detlev W. Hoffmann

Received: May 29, 2001 Revised: November 10, 2001

Communicated by Ulf Rehmann

Abstract. By a theorem of Elman and Lam, fields over which quadratic forms are classified by the classical invariants dimension, signed discriminant, Clifford invariant and signatures are exactly those fields F for which the third power I^{3}F of the fundamental ideal IF in the Witt ring W F is torsion free. We study the possible values of the uinvariant (resp. the Hasse number ũ) of such fields, i.e. the supremum of the dimensions of anisotropic torsion (resp. anisotropic totally indefinite) forms, and we relate these invariants to the symbol length λ, i.e. the smallest integer n such that the class of each product of quaternion algebras in the Brauer group of the field can be represented by the class of a product of ≤ n quaternion algebras. The nonreal case has been treated before by B. Kahn. Here, we treat the real case which turns out to be considerably more involved.

1991 Mathematics Subject Classification: 11E04, 11E10, 11E81, 12D15 Keywords and Phrases: quadratic form, indefinite quadratic form, torsion quadratic form, real field, u-invariant, Hasse number, symbol length

1. Introduction

Let F be a field of characteristic ≠ 2. An important topic in the algebraic theory of quadratic forms over F is the determination of the supremum of the dimensions of certain types of anisotropic quadratic forms over F. For a general survey on this problem, see [H4]. In the present article, we focus on the u-invariant and the Hasse number ũ of F, where u(F) (resp. ũ(F)) is defined as the supremum of the dimensions of anisotropic forms which are torsion in the Witt ring of F (resp. totally indefinite, i.e. indefinite with respect to each ordering on F). By Pfister’s local-global principle, torsion forms are exactly those forms which have signature 0 with respect to each ordering, they are in particular totally indefinite (or t.i. for short). Hence, u(F) ≤ ũ(F). In the absence of orderings, i.e. for nonreal fields, every form is a torsion form and the two definitions coincide with what was originally called the u-invariant, namely the supremum of the dimensions of anisotropic forms over F.

We will relate these two invariants to another one, the so-called symbol length λ, which is defined to be the smallest n (if such an n exists) such that any tensor product of quaternion algebras over F is Brauer-equivalent to a tensor product of ≤ n quaternion algebras. λ(F) ≤ 1 is equivalent to saying that the classes of quaternion algebras form a subgroup of the Brauer group Br(F). In this case, the field is called linked. It should be remarked that by Merkurjev’s theorem [M1], the classes of products of quaternion algebras are exactly the elements in Br_{2}(F), i.e. the elements of exponent ≤ 2 in the Brauer group Br(F).

Perhaps the first result relating the u-invariant and the Hasse number to the symbol length is due to Elman and Lam [EL2], [E] who determined the values of u and ũ for linked fields. Their result reads as follows.

Theorem 1.1. Let F be a linked field. Then u(F) = ũ(F) ∈ {0, 1, 2, 4, 8}. In particular, I_{t}^{4}F = 0. Furthermore, for 1 ≤ n ≤ 3, u(F) = ũ(F) ≤ 2^{n−1} iff I_{t}^{n}F = 0.

In the wake of Merkurjev’s construction of fields with u = 2n for any positive integer n ([M2]) which is based on his index reduction results and its consequences (see Lemma 2.2(iii)) and on a simple fact concerning Albert forms (see Lemma 2.2(i)), it has been noted by Kahn that for nonreal fields, a lower bound for u can easily be given in terms of λ. More precisely, Kahn [Ka, Th. 2] shows the following.

Theorem 1.2. Let F be a nonreal field. Then (i) λ(F) = 0 iff u(F) ≤ 2. (ii) If λ(F) ≥ 1 then u(F) ≥ 2λ(F) + 2. (iii) If λ(F) ≥ 1 and I^{3}F = 0, then u(F) = 2λ(F) + 2.

(In Kahn’s original statement, it was implicitly assumed that λ(F) ≥ 1, and only parts (ii) and (iii) were stated.)

The aim of the present paper is to generalize this result to real fields, in particular to real fields with I_{t}^{3}F = 0. Since the quaternion algebra (−1, −1)_{F} will always be a division algebra over any given real field F, we will always have λ(F) ≥ 1. By Elman and Lam’s theorem 1.1 we know for real F that λ(F) = 1 implies u(F) = ũ(F) ∈ {0, 2, 4, 8} and that in this case u(F) = ũ(F) ∈ {0, 2, 4} iff I_{t}^{3}F = 0. Thus, we are mainly interested in the case F real and λ(F) ≥ 2.

Now fields with I_{t}^{3}F = 0 are also interesting from a different point of view as by another theorem of Elman and Lam [EL3] these are exactly the fields over which quadratic forms can be classified by the classical invariants dimension, signed discriminant, Clifford invariant, and signatures.

Our first main result is the analogue for real fields of Kahn’s theorem above, but now in terms of the Hasse number.

Theorem 1.3. Let F be a real field with λ = λ(F) ≥ 2. Then the following holds.

(i) ũ(F) ≥ 2λ + 2. (ii) If I_{t}^{3}F = 0 and ũ(F) < ∞, then ũ(F) = 2λ + 2.

The situation for the u-invariant seems to be more complicated. We could prove an analogue of Kahn’s theorem only under invoking rather restrictive additional hypotheses on the space of orderings X_{F} of the field. Recall that the reduced stability index st(F) of a real F can be defined as follows : st(F) = 0 if F is uniquely ordered; otherwise, st(F) is the smallest integer s ≥ 0 such that for each basic clopen set H(a_{1}, ⋯, a_{n}) ⊂ X_{F} there exist b_{i} ∈ F^{∗}, 1 ≤ i ≤ s, such that H(a_{1}, ⋯, a_{n}) = H(b_{1}, ⋯, b_{s}). st(F) ≤ 1 is equivalent to F being SAP (cf. [KS, Kap. 3, § 7, Satz 3]).

Theorem 1.4. Let F be a real field with λ = λ(F) ≥ 2.

(i) If st(F) ≤ 1 then u(F) ≥ 2λ. (ii) If I_{t}^{3}F = 0 and st(F) ≤ 2, then u(F) ≤ 2λ + 2.

These results will be shown in the next section.

In [M2], Merkurjev constructed to each n ≥ 1 fields with u(F) = 2n and I^{3}F = 0. It has been shown by Hornix [Hor, Th. 3.5] and Lam [L2] that for each n ≥ 3 there exist real fields F, F′ such that u(F) = ũ(F) = 2n and u(F′) + 2 = ũ(F′) = 2n. Note that in [L2], it was in addition shown that there exist such fields which are uniquely ordered, but nothing was said about I_{t}^{3}F, whereas in [Hor] it was shown that one can construct such fields with I_{t}^{3} = 0, but there were no statements made on the space of orderings of such fields.

For the reader’s convenience, we will give a proof of these results by Hornix resp. Lam in section 3. Our constructions are slightly different from those given by Hornix and Lam but, just as theirs, rely heavily on Merkurjev’s index reduction results as stated in Lemma 2.2. In our constructions, we will also combine the properties of F having I_{t}^{3}F = 0 and of F being uniquely ordered in the case ũ < ∞.

In fact, we will put these results into a larger context where we classify all realizable values for the invariants λ, u and ũ (and their interdependences) for real fields with I_{t}^{3}F = 0 which are SAP. Since the values of u and ũ for fields (real or not) with λ ≤ 1 are covered by Elman and Lam’s theorem 1.1 (note that these fields are always SAP since for them ũ will be finite, [EP, Theorem 2.5]), and since the case of nonreal fields is treated in Kahn’s Theorem 1.2, we will only consider the case of real SAP fields with I_{t}^{3}F = 0 and λ(F) ≥ 2.

Theorem 1.5. Let M = {(n, 2n, 2n + 2), (n, 2n + 2, 2n + 2); n ≥ 2} ∪ {(n, 2n, ∞), (n, 2n + 2, ∞); n ≥ 2} ∪ {(∞, ∞, ∞)}.

(i) Let F be a real SAP field such that λ(F) ≥ 2 and I_{t}^{3}F = 0. Then (λ(F), u(F), ũ(F)) ∈ M.

(ii) Let (λ, u, ũ) ∈ M. Then there exists a real SAP field F with I_{t}^{3}F = 0 and (λ(F), u(F), ũ(F)) = (λ, u, ũ). In the case where ũ < ∞ or λ = ∞, there exist such fields which are uniquely ordered.

As a consequence, we obtain

Corollary 1.6. Let F be a real field with I_{t}^{3}F = 0. Then

(u(F), ũ(F)) ∈ {(2n, 2n); n ≥ 0} ∪ {(2n, ∞); n ≥ 0} ∪ {(2n, 2n + 2); n ≥ 2} .

All pairs of values on the right hand side can be realized as pairs (u(F), ũ(F)) for suitable real F.

As far as notation, terminology and basic results from the algebraic theory of quadratic forms is concerned, we refer to the books by Lam [L1] and Scharlau [S]. In particular, φ ≅ ψ (resp. φ ∼ ψ) denotes isometry (resp. equivalence in the Witt ring) of the forms φ and ψ. ∑ F^{2} denotes all nonzero sums of squares in F. The signed discriminant (resp. Clifford invariant) of a form φ will be denoted by d_{±}φ (resp. c(φ)), and we write φ_{an} for the anisotropic part of φ. An n-fold Pfister form is a form of type ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩, a_{i} ∈ F^{∗}, and we write ⟨⟨a_{1}, ⋯, a_{n}⟩⟩ for short. The set of forms isometric (resp. similar) to n-fold Pfister forms will be denoted by P_{n}F (resp. GP_{n}F). I_{t}^{n}F is the torsion part of I^{n}F, the n-th power of the fundamental ideal IF of classes of evendimensional forms in the Witt ring W F of the field F. The space of orderings of a real field F will be denoted by X_{F}. General references for the SAP property and the reduced stability index are the book by Knebusch and Scheiderer [KS], and the articles [P], [ELP], [EP]. Another property in this context is the socalled ED property (effective diagonalization). It is known that ED implies SAP (but not conversely in general), and that fields with finite ũ have the ED property. Cf. [PW] for more details on ED.

2. Fields with torsion-free I^{3}

Definition 2.1. (i) Let A be a central simple algebra over F (CSA/F) such that its Brauer class [A] is in Br_{2}(F). The symbol length t(A) of A is defined as

t(A) = min{n | ∃ quaternion algebras Q_{i}/F, 1 ≤ i ≤ n, s.t. [A] = [⨂_{i=1}^{n} Q_{i}]} .

(ii) The symbol length λ(F) of the field F is defined as

λ(F) = sup{t(A) | A CSA/F, [A] ∈ Br_{2}(F)} .

(iii) Let φ be a form over F. Let A be a CSA/F such that c(φ) = [A] ∈ Br_{2}(F), where c(φ) denotes the Clifford invariant of φ. Then t(φ) := t(A).

The following lemma compiles some well known results and some special cases of Merkurjev’s index reduction theorem which we will use in this and the following section. We refer to [M2], [T] for details (see also [L1, Sect. 3, Ch. V] for basic results on Clifford invariants and how to compute them).

Lemma 2.2. (i) Let Q_{i} = (a_{i}, b_{i}), 1 ≤ i ≤ n, be quaternion algebras over F with associated norm forms ⟨⟨a_{i}, b_{i}⟩⟩ ∈ P_{2}F. Let A = ⨂_{i=1}^{n} Q_{i} (over F). Then there exist r_{i} ∈ F^{∗}, 1 ≤ i ≤ n, and a form q ∈ I^{2}F, dim q = 2n + 2 such that c(q) = [A] ∈ Br_{2} F and q ∼ ∑_{i=1}^{n} r_{i}⟨⟨a_{i}, b_{i}⟩⟩ in W F. (We will call such a form q an Albert form associated with A.) Furthermore, if t(A) = n (in particular if A is a division algebra), then every Albert form associated with A is anisotropic.

(ii) If q is a form over F with either dim q = 2n + 2 and q ∈ I^{2}F, or dim q = 2n + 1, or dim q = 2n and d_{±}q ≠ 1, then there exist quaternion algebras Q_{i} = (a_{i}, b_{i}), 1 ≤ i ≤ n, such that for A = ⨂_{i=1}^{n} Q_{i} we have c(q) = [A], and there exists an Albert form φ associated with A such that q ⊂ φ.

(iii) If A as in (i) is a division algebra and if ψ is a form over F of one of the following types:

(a) dim ψ ≥ 2n + 3,

(b) dim ψ = 2n + 2 and d_{±}ψ ≠ 1,

(c) dim ψ = 2n + 2, d_{±}ψ = 1 and c(ψ) ≠ [A] ∈ Br_{2} F,

(d) ψ ∈ I^{3}F,

then A stays a division algebra over F (ψ).

The next result will be used in the proofs of Theorem 1.4(ii) and of Lemma 2.4(ii), which in turn will be used in the proof of Theorem 1.3(ii).

Lemma 2.3. Let n ≥ 1 and suppose that I_{t}^{n+1}F = 0. Let φ be a form over F of dimension > 2^{n}. Suppose that either

• φ ∈ I_{t}^{n}F, or

• φ is t.i. and F is ED. If there exists ρ ∈ GP_{n}F such that ρ ⊂ φ, then φ is isotropic.

Proof. Write φ ≅ ρ ⊥ ψ. By assumption, dim ψ ≥ 1. After scaling, we may assume that ρ ∈ P_{n}F. Note that sgn_{P} ρ ∈ {0, 2^{n}} for all P ∈ X_{F}. Let Y = {P ∈ X_{F} | sgn_{P} (ρ) = 2^{n}}.

If ρ is torsion, i.e. if Y is empty, then for any x represented by ψ we have that ρ ⊗ ⟨⟨−x⟩⟩ ∈ P_{n+1}F ∩ W_{t}F ⊂ I_{t}^{n+1}F = 0. Thus, the Pfister neighbor ρ ⊥ ⟨x⟩ is isotropic. Hence, φ is isotropic as it contains ρ ⊥ ⟨x⟩ as subform.

So assume that Y ≠ ∅. First, suppose that φ ∈ I_{t}^{n}F. Then we have sgn_{P} ψ = −2^{n} for all P ∈ Y and hence dim ψ ≥ 2^{n}. Now ⟨1, 1⟩ ⊗ φ ∈ I_{t}^{n+1}F = 0, hence ⟨1, 1⟩ ⊗ ρ ∼ −⟨1, 1⟩ ⊗ ψ in W F. By β-decomposition (cf. [EL1, p. 289]), we can write ψ ≅ γ ⊥ σ with ⟨1, 1⟩ ⊗ γ ∼ 0 (in particular, γ ∈ W_{t}F), dim σ = dim ρ = 2^{n} and ⟨1, 1⟩ ⊗ ρ ≅ −⟨1, 1⟩ ⊗ σ. Comparing signatures, we see that sgn_{P} ρ = − sgn_{P} σ ∈ {0, 2^{n}}. Now let x ∈ F^{∗} be any element represented by σ. The above shows that x <_{P} 0 for all P ∈ Y. For all other P ∈ X_{F}, ρ is indefinite. This yields that ρ ⊥ ⟨x⟩ is t.i. and a Pfister neighbor of ρ ⊗ ⟨⟨−x⟩⟩ which is therefore torsion. We conclude as before that φ is isotropic.

Finally, suppose that φ is t.i. and that F is ED. Since ρ is positive definite at all orderings P ∈ Y, and since φ ≅ ρ ⊥ ψ is t.i., ED implies that ψ represents an x ∈ F^{∗} such that x <_{P} 0 for all P ∈ Y. Then ρ ⊥ ⟨x⟩ is t.i. and a Pfister neighbor contained in φ, and we conclude as before that φ is isotropic.

For later purposes, we now state some useful facts on u and ũ of real fields with I_{t}^{3}F = 0.

Lemma 2.4. Let F be a field with I_{t}^{3}F = 0. Then the following holds.

(i) If 2 < u(F) < ∞, then there exists an anisotropic form φ ∈ I_{t}^{2}F such that dim φ = u(F).

(ii) If ũ(F) < ∞, then ũ(F) is even. Furthermore, if 2 < ũ(F) < ∞, then there exists an anisotropic t.i. form φ ∈ I^{2}F such that dim φ = ũ(F) and sgn_{P} (φ) ∈ {0, 4} for all P ∈ X_{F}.

Proof. (i) See [EL1, Prop. 1.4].

(ii) See [ELP, Th. H] for a proof that ũ(F) is even if it is finite. Now suppose φ is anisotropic, t.i. and dim φ = ũ(F) ≥ 4. Since ũ(F) is finite, F has ED and one easily sees that φ contains a 3-dimensional t.i. subform τ′. Then τ′ is a Pfister neighbor of some anisotropic torsion τ ∈ P_{2}F. Thus, if ũ(F) = 4, this τ is the desired form. So we may assume that ũ(F) ≥ 6.

Since F is SAP, we may scale φ so that sgn_{P} φ ≥ 0 for all P ∈ X_{F}. Consider the clopen set Y = {P ∈ X_{F} | sgn_{P} φ ≥ 5}. Since F is SAP, there exists a 3-fold Pfister form π such that sgn_{P} π = 8 for all P ∈ Y and sgn_{P} π = 0 otherwise. Consider φ_{1} = x(φ ⊥ −π)_{an}, where x ∈ F^{∗} is chosen so that sgn_{P} φ_{1} ≥ 0 for all P ∈ X_{F}. By construction, 0 ≤ sgn_{P} φ_{1} ≤ max{4, | sgn_{P} φ − 8|} < dim φ. If dim φ_{1} > dim φ, then φ_{1} would be an anisotropic t.i. form of dimension ≥ ũ(F)+2, clearly a contradiction. If dim φ_{1} < dim φ, then φ and π would contain a common 5-dimensional subform which, being a Pfister neighbor, would in turn contain a subform ρ ∈ GP_{2}F. Since F is ED as ũ(F) < ∞, Lemma 2.3 then implies that φ is isotropic, a contradiction. It follows that dim φ_{1} = dim φ. By repeating this construction, we get a sequence of anisotropic t.i. forms φ_{0} = φ, φ_{1}, ⋯, φ_{r} such that for i ≥ 1 we have dim φ_{i} = dim φ, 0 ≤ sgn_{P} φ_{i} ≤ max{4, | sgn_{P} φ_{i−1} − 8|} and 0 ≤ sgn_{P} φ_{r} ≤ 4 for all P ∈ X_{F}.

Hence, we may assume that φ is anisotropic t.i., dim φ = ũ(F) and 0 ≤ sgn_{P} φ ≤ 4 for all P ∈ X_{F}. Let d = d_{±}φ and consider ψ = (φ ⊥ ⟨1, −d⟩)_{an}. Note that ψ ∈ I^{2}F and therefore sgn_{P} ψ ≡ 0 mod 4. Since 0 ≤ sgn_{P} φ ≤ 4 and sgn_{P} ⟨1, −d⟩ ∈ {0, ±2} for all P ∈ X_{F}, it follows readily that sgn_{P} ψ ∈ {0, 4}. We also have that dim φ − 2 ≤ dim ψ ≤ dim φ + 2.

If dim ψ = dim φ + 2, then ψ ≅ φ ⊥ ⟨1, −d⟩ would be an anisotropic t.i. form of dimension ũ(F) + 2, clearly a contradiction.

If dim ψ = dim φ − 2, then φ ≅ ψ ⊥ ⟨d, −1⟩. Since sgn_{P} ψ ≥ 0 for all P ∈ X_{F} and because of ED, we have that ψ represents some a ∈ ∑ F^{2}. Then ψ ⊥ −aψ is a torsion form in I^{3}F and thus hyperbolic. But ψ ⊥ −aψ contains the subform ψ ⊥ ⟨−1⟩ which by dimension count must be isotropic. Hence φ is isotropic, a contradiction. Thus dim ψ = dim φ = ũ(F) and ψ is the desired form.

Remark 2.5. (i) If u(F) = ∞, then there exist anisotropic torsion forms in I^{2}F of arbitrarily large dimension. Indeed, let φ ∈ W_{t}F be anisotropic of dimension ≥ 2n + 2. Let d = d_{±}φ and consider ψ = (φ ⊥ ⟨1, −d⟩)_{an}. Then one readily checks that dim ψ ≥ 2n and ψ ∈ I_{t}^{2}F.

(ii) If ũ(F) = ∞, then there exist anisotropic t.i. forms in I^{2}F of arbitrarily large dimension. Indeed, let φ be any anisotropic t.i. form of dimension 4n + 3 for any n ≥ 1 (such φ exists by [ELP, Th. A]). Let d be such that φ ⊥ ⟨d⟩ ∈ I^{2}F. Let ψ = (φ ⊥ ⟨d⟩)_{an}. Then ψ ∈ I^{2}F and dim ψ ∈ {4n + 2, 4n + 4}. If dim ψ = 4n + 4 then ψ ≅ φ ⊥ ⟨d⟩ is t.i.. If dim ψ = 4n + 2, then sgn_{P} ψ ≡ 0 mod 4 for all P ∈ X_{F} as ψ ∈ I^{2}F, and therefore | sgn_{P} ψ| ≤ 4n < 4n + 2 = dim ψ for all P ∈ X_{F}. Again, ψ is t.i..

Let us now turn to the proof of part (ii) of Theorem 1.4 where we assume that I_{t}^{3}F = 0 and st(F) ≤ 2. In [KS, Kap. 3, § 7, Korollar], one finds different characterizations of F having reduced stability index ≤ s for an integer s ≥ 1. The one we are interested in is the following : st(F) ≤ s is equivalent to (I^{s+1}F)_{red} = 2(I^{s}F)_{red}, i.e. for each form φ ∈ I^{s+1}F there exists a form ψ ∈ I^{s}F such that sgn_{P} φ = sgn_{P} (⟨1, 1⟩ ⊗ ψ) for all P ∈ X_{F}. If I_{t}^{s+1}F = 0, then st(F) ≤ s is therefore equivalent to I^{s+1}F = 2I^{s}F. By [Kr, Prop. 1], we thus get

Lemma 2.6. Let s ≥ 1 be an integer and let F be a real field with I_{t}^{s+1}F = 0. Then the following are equivalent :

(i) st(F) ≤ s; (ii) I^{s+1}F = 2I^{s}F; (iii) I^{s+1}F (\sqrt{−1}) = 0.

Now I^{s+1}F (\sqrt{−1}) = 0 implies I_{t}^{s+1}F = 0, [Kr, Prop. 1], and in view of this lemma, we may replace the hypotheses I_{t}^{3}F = 0 plus st(F) ≤ 2 by I^{3}F (\sqrt{−1}) = 0. We then get the following result which holds for any field (not just for real fields) and which implies the second part of Theorem 1.4.

Theorem 2.7. Suppose that I^{3}F (\sqrt{−1}) = 0. Then

u(F) ≤ min\{4λ(F (\sqrt{−1})) + 2, 2λ(F) + 2\} .

Proof. First, we prove that u(F) ≤ 2λ(F) + 2. If the level s(F) of F is finite, i.e. F is nonreal, then this follows from Kahn’s theorem 1.2.

So assume that F is a real field with I^{3}F (\sqrt{−1}) = 0. We will show that if φ ∈ I_{t}^{2}F with t(φ) = t, then dim φ > 2t + 2 implies that φ is isotropic. This then implies readily u(F) ≤ 2λ(F) + 2. Indeed, this follows from the fact that there always exists an anisotropic form in I_{t}^{2}F of dimension u(F) if u(F) is finite (Lemma 2.4(i)), resp. of arbitrarily large dimension if u(F) is infinite (Remark 2.5(i)), and the fact that in the case of a real F with I_{t}^{3}F = 0, st(F) ≤ 2 is equivalent to I^{3}F (\sqrt{−1}) = 0 by Lemma 2.6.

Now let φ ∈ I_{t}^{2}F with t(φ) = t and dim φ > 2t + 2. We will prove by induction on t that φ is isotropic. If t = 0 then φ ∈ I_{t}^{3}F = 0 and φ is in fact hyperbolic. If t = 1 then there exists (an anisotropic) τ ∈ P_{2}F such that c(φ) = c(τ). By Merkurjev’s theorem, φ ≡ τ mod I^{3}F. Since sgn_{P} τ ∈ {0, 4} and 0 = sgn_{P} φ ≡ sgn_{P} τ mod 8 for all P ∈ X_{F}, we see that τ ∈ W_{t}F, hence φ ≡ τ mod I_{t}^{3}F and thus φ ∼ τ ∈ W F as I_{t}^{3}F = 0. Hence dim φ > dim φ_{an} = dim τ = 4 and φ is isotropic.

Let t ≥ 2. By Lemma 2.2(i), there exists an anisotropic (2t+2)-dimensional form τ ∈ I^{2}F such that φ ≡ τ mod I^{3}F. Let d ∈ F^{∗} such that τ_{F (\sqrt{d})} is isotropic. Let τ′ ∈ I^{2}F (\sqrt{d}) such that (τ_{F (\sqrt{d})})_{an} ≅ τ′. Then dim τ′ ≤ 2t. Hence, t(τ′) ≤ t−1. (In fact, one can readily show that dim τ′ = 2t and t(τ′) = t − 1, but we won’t need this here.) Also, φ_{F (\sqrt{d})} ≡ τ′ mod I^{3}F (\sqrt{d}). By [EL3, Th. 3] and [EL4, Cor. 4.6], we have that I_{t}^{3}F (\sqrt{d}) = 0 and I^{3}F (\sqrt{d})(\sqrt{−1}) = 0. By induction hypothesis, we have dim(φ_{F (\sqrt{d})})_{an} ≤ 2(t − 1) + 2 = 2t. Now dim φ ≥ 2t + 4, hence there exist a, b ∈ F^{∗} such that ⟨a, b⟩ ⊗ ⟨1, −d⟩ ⊂ φ (cf. [L1, Ch. VII, Lemma 3.1]). Now ⟨a, b⟩ ⊗ ⟨1, −d⟩ ∈ GP_{2}F, and by Lemma 2.3, φ is isotropic.

Let us now show that u(F) ≤ 4λ(F (\sqrt{−1})) + 2. This is trivially true for s(F) = 1 as in this case we have F = F (\sqrt{−1}) and already u(F) ≤ 2λ(F) + 2.

So suppose that s(F) ≥ 2. We put L = F (\sqrt{−1}) and we may assume that λ = λ(L) < ∞. Since I_{t}^{3}F = 0, we have ⟨1, 1⟩I_{t}^{2}F = 0. Hence, ann(⟨1, 1⟩) ∩ I^{2}F = ann(⟨1, 1⟩) ∩ I_{t}^{2}F = I_{t}^{2}F. Consider the Scharlau transfer s_{∗} : W L → W F induced by the F -linear map L → F defined by 1 ↦ 0 and \sqrt{−1} ↦ 1. Note that for any form ρ over L there exists a form σ over F such that dim σ ≤ 2 dim ρ and s_{∗}(ρ) ∼ σ in W F.

By [AEJ, Prop. 1.24], we have s_{∗}(I^{2}L) = ann(⟨1, 1⟩) ∩ I^{2}F and thus s_{∗}(I^{2}L) = I_{t}^{2}F. Now let ψ be any form in I^{2}L. By Lemma 2.2(i), there exists a form η ∈ I^{2}L such that dim η ≤ 2λ + 2 and c(ψ) = c(η) ∈ Br_{2} L. After scaling, we may assume that η ≅ ⟨1⟩ ⊥ η′. In particular, there exists a form γ ∈ I^{3}L such that η ∼ ψ + γ in W L. Now s_{∗}(γ) ∈ I_{t}^{3}F = 0. Hence s_{∗}(ψ) = s_{∗}(η) = s_{∗}(⟨1⟩) + s_{∗}(η′) ∼ σ for some form σ over F with dim σ ≤ 2 dim η′ ≤ 4λ + 2.

Now let φ ∈ I_{t}^{2}F. Since s_{∗}(I^{2}L) = I_{t}^{2}F, the above shows that φ ∼ μ in W F for some form μ over F with dim μ ≤ 4λ + 2. Hence, if φ is anisotropic we necessarily have dim φ ≤ 4λ + 2.

Suppose u(F) = ∞. Then there exists some anisotropic form τ ∈ W_{t}F with dim τ ≥ 4λ + 6 and dim τ even. Let d = d_{±}τ. Then one easily sees that τ ⊥ ⟨1, −d⟩ ∈ I_{t}^{2}F, and its anisotropic part must therefore be of dimension ≤ 4λ + 2, a contradiction to τ being anisotropic and dim τ ≥ 4λ + 6.

Hence u(F) < ∞. Then Lemma 2.4(i) and the above imply that u(F) ≤ 4λ + 2.

Remark 2.8. Let F be such that s(F) ≥ 2 and let L = F (\sqrt{−1}). Define u′(F) = sup{dim φ | φ anisotropic form/F and ⟨1, 1⟩ ⊗ φ = 0 ∈ W F }. It was shown in [Pf, Ch. 8, Th. 2,12] that u′(F) ≤ 2u(L) − 2. Now if I_{t}^{3}F = 0, then one readily verifies that u(F) = u′(F) (see also [Pf, Ch. 8, Prop. 2.6]). Hence, this would imply that u(F) ≤ 2u(L) − 2. Note, however, that I^{3}L need not be zero and that therefore u(L) > 2λ(L) + 2 might very well be possible (cf. Theorem 1.2), in which case our bound u(F) ≤ 4λ(L) + 2 would be better.

Corollary 2.9. (See also [Pf, Ch. 8, Th. 2,12], [EL1, Th. 4.11].) Let F be a field with s(F) ≥ 2 and let L = F (\sqrt{−1}). Let n ∈ \{1, 2, 3\}. Then u(L) ≤ 2n implies u(F) ≤ 4n−2. Furthermore, if u(L) = 1 then F is real and pythagorean (i.e. u(F) = 0).

Proof. If u(L) ≤ 2n, 1 ≤ n ≤ 3, then I^{3}L = 0 and thus I_{t}^{3}F = 0 (cf. [Kr, Prop. 1]). Theorem 1.2 yields λ(L) ≤ n − 1. Hence u(F) ≤ 4λ(L) + 2 ≤ 4n − 2. The second part is left to the reader.

To prove Theorems 1.3 and 1.4(i), we will need the following lemma.

Lemma 2.10. Let n ≥ 1 and suppose that F is SAP.

(i) Let π_{i} ∈ P_{n}F, 1 ≤ i ≤ r. Then there exists a form φ ∈ I^{n}F such that sgn_{P} φ ∈ {0, 2^{n}} for all P ∈ X_{F}, and φ ≡ ∑_{i=1}^{r} π_{i} mod I^{n+1}F.

(ii) If I_{t}^{n+1}F = 0, and if φ ∈ I^{n}F such that sgn_{P} φ ∈ {0, 2^{n}} for all P ∈ X_{F},

then φ ≅ φ_{t} ⊥ φ_{0} with φ_{t} ∈ W_{t}F and dim φ_{0} ∈ {0, 2^{n}}. (iii) If I_{t}^{n+1}F = 0, then the form φ in part (i) can be chosen so as to have

dimension ≤ r2^{n} − 2r + 2.

Proof. (i) We use induction on r. If r = 1 then φ = π_{1} will do. So suppose r ≥ 2. By induction hypothesis, there exists a form ψ such that ψ ≡ ∑_{i=1}^{r−1} π_{i} mod I^{n+1}F and sgn_{P} ψ ∈ {0, 2^{n}} for all P ∈ X_{F}. Let φ̂ = ψ ⊥ −π_{r}. Since sgn_{P} π_{r} ∈ {0, 2^{n}}, we have sgn_{P} φ̂ ∈ {0, ±2^{n}}. Since F is SAP, there exists an x ∈ F^{∗} such that φ = xφ̂ has sgn_{P} φ ∈ {0, 2^{n}} for all P ∈ X_{F}. Clearly, φ ≡ ∑_{i=1}^{r} π_{i} mod I^{n+1}F.

(ii) Suppose now that I_{t}^{n+1}F = 0. Consider the clopen set Y = {P ∈ X_{F} | sgn_{P} φ = 2^{n}} in X_{F}. If Y is empty then φ ∈ W_{t} and there is nothing to show. So suppose Y ≠ ∅. Let σ ∈ P_{n}F be such that sgn_{P} σ = 2^{n} if P ∈ Y, and sgn_{P} σ = 0 otherwise. Such σ exists as F is SAP. It follows that ⟨1, 1⟩ ⊗ φ ≡ ⟨1, 1⟩ ⊗ σ mod I_{t}^{n+1}F (both forms are in I_{t}^{n+1}F and have the same signatures). Now I_{t}^{n+1}F = 0 and thus ⟨1, 1⟩ ⊗ φ ∼ ⟨1, 1⟩ ⊗ σ. (Note that ⟨1, 1⟩ ⊗ σ is anisotropic because sgn_{P} ⟨1, 1⟩ ⊗ σ = dim ⟨1, 1⟩ ⊗ σ = 2^{n+1} for all P ∈ Y ≠ ∅.) Comparing dimensions and using β-decomposition (cf. [EL1, p. 289]), we see that φ ≅ φ_{t} ⊥ φ_{0} with φ_{t} ∈ W_{t}F and dim φ_{0} = dim σ = 2^{n}.

(iii) We use a similar induction argument as in (i), but we assume in addition that the form ψ there is of dimension ≤ (r −1)2^{n} −2(r −1)+2. By (ii), we can write ψ ≅ ψ_{t} ⊥ ψ_{0} with dim ψ_{0} ∈ {0, 2^{n}}, ψ_{t} ∈ W_{t}F, and dim ψ_{0} = 2^{n} only if there exists some P ∈ X_{F} with sgn_{P} ψ = 2^{n}. Let y ∈ D(ψ_{0}) if dim ψ_{0} = 2^{n}, and let y ∈ D(ψ) otherwise. One readily checks that sgn_{P} yψ = sgn_{P} ψ ∈ {0, 2^{n}} and that yψ ≅ ⟨1⟩ ⊥ ψ′. Let now π_{r} ≅ ⟨1⟩ ⊥ π′_{r} and let φ′ = ψ′ ⊥ −π′_{r}. Note that dim φ′ ≤ r2^{n} − 2r + 2. As in the proof of (i), sgn_{P} φ′ ∈ {0, ±2^{n}}, and after scaling, we obtain the form φ with sgn_{P} φ ∈ {0, 2^{n}} for all P ∈ X_{F}, dim φ = dim φ′ ≤ r2^{n} − 2r + 2, and φ ≡ ∑_{i=1}^{r} π_{i} mod I^{n+1}F.

Proof of Theorem 1.3. (i) If F is not SAP, then ũ(F) = ∞ and there is nothing to show. So suppose that F is SAP. Let A = Q_{1} ⊗ ⋯ ⊗ Q_{t} ∈ Br_{2}F, where the Q_{i} are quaternion algebras such that t(A) = t ≥ 2, and consider the norm forms π_{i} ∈ P_{2}F associated with Q_{i}. By Lemma 2.10(i), there exists an anisotropic form φ ∈ I^{2}F such that φ ≡ ∑_{i=1}^{t} π_{i} mod I^{3}F and sgn_{P} φ ∈ {0, 4}. Note that c(φ) = [A]. If dim φ ≤ 2t then, by Lemma 2.2(ii), c(φ) could be represented by a product of fewer than t quaternion algebras, a contradiction to t(A) = t. Hence dim φ ≥ 2t + 2. Note that φ is t.i. provided t ≥ 2.

If λ(F) = ∞, then for any t ≥ 1 there exists an A ∈ Br_{2}F with t(A) = t, and the above shows that ũ(F) = ∞. If λ(F) < ∞, then choose A as above such that t(A) = λ(F). The above shows that ũ(F) ≥ 2λ(F) + 2.

(ii) By Lemma 2.4(ii), we may assume that there exists an anisotropic t.i. form φ ∈ I^{2}F with dim φ = ũ(F) and sgn_{P} φ ∈ {0, 4} for all P ∈ X_{F}. Let t(φ) = t ≤ λ and let c(φ) = Q_{1} ⊗ ⋯ ⊗ Q_{t} ∈ Br_{2}F. With π_{i} the norm forms associated with Q_{i}, we get φ ≡ ∑_{i=1}^{t} π_{i} mod I^{3}F.

By Lemma 2.10(iii), there exists a form ψ ∈ I^{2}F, dim ψ ≤ 2t + 2 such that sgn_{P} ψ ∈ {0, 4} for all P ∈ X_{F} and such that φ ≡ ψ mod I^{3}F. Since sgn_{P} φ ≡ sgn_{P} ψ mod 8, this readily yields φ ⊥ −ψ ∈ I_{t}^{3}F = 0. The anisotropy of φ then shows that ũ(F) = dim φ ≤ 2t + 2 ≤ 2λ(F) + 2, which together with (i) yields ũ(F) = 2λ(F) + 2.

Proof of Theorem 1.4(i). Let A = Q_{1} ⊗ ⋯ ⊗ Q_{t} ∈ Br_{2}F, where the Q_{i} are quaternion algebras such that t(A) = t ≥ 2. As in part (i) of the proof of Theorem 1.3, there exists an anisotropic form φ ∈ I^{2}F such that c(φ) = [A], sgn_{P} φ ∈ {0, 4}, dim φ ≥ 2t + 2.

Now let π ∈ P_{2}F be such that sgn_{P} φ = sgn_{P} π for all P ∈ X_{F}. (Such π exists as F is SAP and sgn_{P} φ ∈ {0, 4}.) Consider ψ = (φ ⊥ −π)_{an}. By construction, ψ ∈ I_{t}^{2}F and dim ψ ≥ dim φ − 4 = 2t − 2. Suppose that dim ψ = dim φ − 4. Then φ ≅ ψ ⊥ π and we have ψ, π ∈ I^{2}F, c(φ) = c(ψ)c(π). By dimension count and Lemma 2.2(ii), we have t(ψ) ≤ t − 2, t(π) ≤ 1, and therefore t(φ) = t(A) = t ≤ t(ψ) + t(π) ≤ t − 1, a contradiction. Hence, dim ψ ≥ dim φ − 2 = 2t.

If λ(F) = ∞, then for any t ≥ 1 there exists an A ∈ Br_{2}F with t(A) = t, and the above shows that u(F) = ∞.

If λ(F) < ∞, then choose A as above such that t(A) = λ(F). The above then shows that u(F) ≥ 2λ(F).

Since fields with finite ũ are always SAP, the following is an immediate consequence of Theorems 1.3, 1.4.

Corollary 2.11. Let F be a real field with I_{t}^{3}F = 0 and ũ(F) < ∞. Then ũ(F) = 2λ(F) + 2 ∈ {u(F), u(F) + 2}.

Example 2.12. The condition in Theorem 1.4(i) that F be SAP seems to be quite restrictive. However, we will certainly need some sort of additional assumption on F besides I_{t}^{3}F = 0 to get the lower bound u(F) ≥ 2λ(F). To see what can go wrong when one drops the assumption that F is SAP, consider the following example. Let F = ℝ((t_{1})) ⋯ ((t_{n})) be the iterated power series field in n variables over the reals. Then, by Springer’s theorem, u(F) = 0. In particlar, I_{t}^{3}F = 0. For n ≥ 2, F is not SAP as for example ⟨1, t_{1}, t_{2}, −t_{1}t_{2}⟩ is not weakly isotropic. However, one can show that λ(F) = [n/2] + 1. The value t(A) = [n/2] + 1 can be realized, for example, by the multiquaternion division algebra A = (−1, −1) ⊗ (t_{1}, t_{2}) ⊗ ⋯ (t_{m−1}, t_{m}) where m = [n/2] (i.e. n ∈ {2m, 2m + 1}).

As for the upper bound for u(F) for a field with I_{t}^{3}F = 0, we proved in Theorem 1.4 that u(F) ≤ 2λ(F) + 2 under the assumption that st(F) ≤ 2. We believe that this additional assumption is in fact superfluous, but we were unable to get this upper bound without it.

Conjecture 2.13. Let F be real with I_{t}^{3}F = 0. Then u(F) ≤ 2λ(F) + 2.

In support of this conjecture, we can prove that it holds for small values of λ(F).

Proposition 2.14. Let F be real with I_{t}^{3}F = 0. If λ = λ(F) ≤ 4 then u(F) ≤ 2λ + 2.

Proof. We will show that if φ is an anisotropic form in I_{t}^{2}F with 1 ≤ t(φ) = t ≤ 4, then dim φ ≤ 2t + 2 and thus dim φ = 2t + 2 by Lemma 2.2(ii), which by Lemma 2.4(i) immediately yields the desired result. (Note that t(φ) = 0 implies that φ ∈ I_{t}^{3}F = 0, i.e. φ is hyperbolic.)

So let φ ∈ I_{t}^{2}F and suppose that 1 ≤ t(φ) = t ≤ 4 and dim φ ≥ 2t + 4. By Lemma 2.2(i), there exists a form ψ ∈ I^{2}F with dim ψ = 2t + 2 such that φ ≡ ψ mod I^{3}F. Now ⟨1, 1⟩ ⊗ φ ∈ I_{t}^{3}F = 0 and ⟨1, 1⟩ ⊗ (φ ⊥ −ψ) ∈ I^{4}F, hence ⟨1, 1⟩ ⊗ ψ ∈ I^{4}F. We have dim ⟨1, 1⟩ ⊗ ψ = 4t + 4 ≤ 20. By the Arason-Pfister Hauptsatz and [H1, Main Theorem], there exists ρ ∈ GP_{4}F such that ⟨1, 1⟩ ⊗ ψ ∼ ρ in W F. After scaling, we may assume that ρ ∈ P_{4}F. Since ρ is divisible by ⟨1, 1⟩, there exists σ ∈ P_{3}F such that ρ ≅ ⟨1, 1⟩ ⊗ σ. Comparing signatures, we see that sgn_{P} ψ = sgn_{P} σ for all P ∈ X_{F}. Thus, φ ⊥ −ψ ⊥ σ ∈ I_{t}^{3}F = 0. Thus, in W F we get φ ⊥ σ ∼ ψ. Now dim(φ ⊥ σ) ≥ 2t + 12 and dim ψ = 2t + 2, hence i_{W} (φ ⊥ σ) ≥ 5. Therefore, φ contains a 5-dimensional Pfister neighbor of σ. Since 5-dimensional Pfister neighbors always contain a subform in GP_{2}F, we have that there exists τ ∈ GP_{2}F such that τ ⊂ φ. Thus, φ is isotropic by Lemma 2.3.

3. Construction of fields with prescribed invariants

We will now focus on the realizability of given triples (λ, u, ũ) for nonlinked SAP-fields with I_{t}^{3} = 0. Let us restate the corresponding theorem from the introduction, whose proof will take up most of the remainder of this section.

Theorem 3.1. Let M = {(n, 2n, 2n + 2), (n, 2n + 2, 2n + 2); n ≥ 2} ∪ {(n, 2n, ∞), (n, 2n + 2, ∞); n ≥ 2} ∪ {(∞, ∞, ∞)}.

(i) Let F be a real SAP field such that λ(F) ≥ 2 and I_{t}^{3}F = 0. Then (λ(F), u(F), ũ(F)) ∈ M.

(ii) Let (λ, u, ũ) ∈ M. Then there exists a real SAP field F with I_{t}^{3}F = 0 and (λ(F), u(F), ũ(F)) = (λ, u, ũ). In the case where ũ < ∞ or λ = ∞, there exist such fields which are uniquely ordered.

Proof. (i) This follows immediately from Theorems 1.3, 1.4.

(ii) We fix once and for all a real field F_{0}. Our constructions will be divided into three cases : Finite λ and finite ũ, finite λ and infinite ũ, and infinite λ.

The case 2 ≤ λ < ∞ and ũ < ∞

Put n = λ + 1. We have to construct fields F, F′ with (λ(F), u(F), ũ(F)) = (n − 1, 2n, 2n) and (λ(F′), u(F′), ũ(F′)) = (n − 1, 2n − 2, 2n).

Let F_{1} = F_{0}(x_{1}, x_{2}, ⋯, y_{1}, y_{2}, ⋯) be the rational function field in an infinite number of variables x_{i}, y_{j} over F_{0}. Consider the multiquaternion algebras A_{n} = (1 + x_{1}^{2}, y_{1}) ⊗ ⋯ ⊗ (1 + x_{n−1}^{2}, y_{n−1}) and B_{n} = A_{n−1} ⊗ (−1, −1), n ≥ 2, which are division algebras (cf. [H2, Lemma 2(iv)]). Let ψ_{n} be a 2n-dimensional Albert form of A_{n} such that ψ_{n} ∼ ∑_{i=1}^{n−1} c_{i}⟨⟨1 + x_{i−1}^{2}, y_{i−1}⟩⟩ in W F_{1} for suitable c_{i} ∈ F_{1}^{∗}, and let ψ′_{n} be a 2n-dimensional Albert form of B_{n} such that ψ′_{n} ∼ ⟨⟨−1, −1⟩⟩ + cψ_{n−1} for suitable c ∈ F_{1}^{∗}. Since sgn_{P} ⟨⟨1 + x_{i−1}^{2}, y_{i−1}⟩⟩ = 0 and sgn_{P} ⟨⟨−1, −1⟩⟩ = 4 for each P ∈ X_{F_{1}}, we have sgn_{P} ψ_{n} = 0 and sgn_{P} ψ′_{n} = 4 for all P ∈ X_{F_{1}}. Now fix any ordering P_{1} ∈ X_{F_{1}}.

Suppose that L is a field such that (A_{n})_{L} (resp. (B_{n})_{L}) is a division algebra and such that P_{1} extends to an ordering P ∈ X_{L}. Consider the following classes of forms over L :

C_{1}(L) = {α | α form/L, dim α = 2n + 1, α indefinite at P }

C_{2}(L) = {α | α form/L, α ∈ I^{3}L, sgn_{P} α = 0}
C_{3}(L) = {α | α form/L, dim α = 2n, sgn_{P} α = 0}

We construct an infinite tower of fields F_{1} ⊂ F_{2} ⊂ ⋯ and F_{1} = F′_{1} ⊂ F′_{2} ⊂ ⋯ as follows. Suppose we have constructed F_{i} (resp. F′_{i}), i ≥ 1 such that (A_{n})_{F_{i}} (resp. (B_{n})_{F′_{i}}) are division algebras and such that P_{1} extends to an ordering P_{i} ∈ X_{F_{i}} (resp. X′_{F_{i}}).

Let F_{i+1} (resp. F′_{i+1}) be the compositum of all function fields F_{i}(α) (resp. F′_{i}(α)) where α ∈ C_{1}(F_{i}) ∪ C_{2}(F_{i}) (resp. C_{1}(F′_{i}) ∪ C_{2}(F′_{i}) ∪ C_{3}(F′_{i})).

Since an ordering P of a field L extends to an ordering of the function field L(α) of a form α over L if and only if α is indefinite at P, we see that there exists an ordering on F_{i+1} (resp. F′_{i+1}) extending the ordering P_{i} since we only take function fields of forms in the C_{i}, and all these forms are indefinite at P_{i} (cf. [ELW, Th. 3.5 and Rem. 3.6]). We will fix such an ordering and call it P_{i+1}. Note that no other ordering on F_{i} (resp. F′_{i}) will extend to F_{i+1} (resp. F′_{i+1}). Indeed, let Q be any ordering on F_{i+1} (resp. F′_{i+1}) and let b ∈ F_{i}^{∗} (resp. F_{i}^{′∗}) be such that b <_{P_{i}} 0 and b >_{Q} 0. Then 2n × ⟨1⟩ ⊥ ⟨b⟩ is in C_{1} and definite at Q, which shows that Q will not extend.

Next, we show that A_{n} (resp. B_{n}) stays a division algebra over F_{i+1} (resp. F′_{i+1}). If α ∈ C_{1}(L)∪C_{2}(L) and A_{n} (resp. B_{n}) is division over L, then it follows immediately from Lemma 2.2(iii), parts (a) and (d) that A_{n} (resp. B_{n}) stays division over L(α). In particular, this shows that (A_{n})_{F_{i+1}} will be division.

To show that B_{n} stays a division algebra over F′_{i+1}, it remains to show that if P_{1} extends to an ordering P on L and B_{n} is division over L, then B_{n} stays a division algebra over L(α) for α ∈ C_{3}(L). If d_{±}α ≠ 1, this follows from Lemma 2.2(iii), part (b). If d_{±}α = 1, then α ∈ I^{2}L, and by Lemma 2.2(iii), part (c) it suffices to show that c(α) ≠ [(B_{n})_{L}] in Br_{2} L. Suppose c(α) = [(B_{n})_{L}]. Since c((ψ′_{n})_{L}) = [(B_{n})_{L}], we have by Merkurjev’s theorem that α ≡ (ψ′_{n})_{L} mod I^{3}L and hence 0 = sgn_{P} α ≡ sgn_{P} (ψ′_{n})_{L} ≡ 4 mod 8, clearly a contradiction.

With the F_{i} and their orderings P_{i} constructed for all i, we now put F = ⋃_{i=1}^{∞} F_{i} (resp. F′ = ⋃_{i=1}^{∞} F′_{i}) and P = ⋃_{i=1}^{∞} P_{i}. P will then be the unique ordering on F (resp. F′) (see also the proof of [H3, Th. 2]). It is also obvious from our construction that I_{t}^{3}F = 0 and that indefinite forms of dimension 2n + 1 will be isotropic. The latter implies by [ELP, Th. A] that ũ(F), ũ(F′) ≤ 2n. Also, A_{n} (resp. B_{n}) will stay a division algebra over F (resp. F′). In the case of F, this means that the form (ψ_{n})_{F} will be a 2n-dimensional torsion form which is anisotropic by Lemma 2.2(i). Hence u(F) ≥ 2n and thus u(F) = ũ(F) = 2n. In the case of F′, we have by a similar reasoning that (ψ′_{n})_{F′} is a 2n-dimensional indefinite anisotropic form (recall that dim(ψ′_{n})_{F′} = 2n ≥ 6 > 4 = sgn_{P} (ψ′_{n})_{F′}). Hence ũ(F′) = 2n. However, by construction, torsion forms of dimension 2n will be isotropic and thus u(F′) ≤ 2n − 2. On the other hand, B_{n} = A_{n−1} ⊗ (−1, −1) will stay a division algebra over F′ and thus also A_{n−1}. Hence, just as before, we will now have the anisotropic (2n − 2)-dimensional torsion form (ψ_{n−1})_{F′}, which shows that u(F′) = 2n − 2.

The fact that λ(F) = λ(F′) = n − 1 follows from Corollary 2.11.

The case 2 ≤ λ < ∞ and ũ = ∞

With F_{0} as above, we let now F_{1} = F_{0}(x_{1}, x_{2}, ⋯, y_{1}, y_{2}, ⋯)((t)), but we keep the definitions of A_{n}, B_{n}, ψ_{n}, ψ′_{n} from above. Let L be any extension of F_{1} such that all orderings of F_{1} extend to L and such that A_{n} (resp. B_{n}) is division over L. This time, we consider the following classes of quadratic forms, where n = λ + 1 ≥ 3.

C_{1}(L) = {α | α form/L, dim α ≥ 2n + 2,
α ≅ α_{0} ⊥ α_{t}, α_{t} ∈ W_{t}L, dim α_{0} ∈ {0, 4}}
C_{2}(L) = {α | α = ⟨1, 1⟩ ⊗ ⟨1, x, y, −xy⟩, x, y ∈ L^{∗}}
C_{3}(L) = {α | α form/L, α ∈ I_{t}^{3}L}
C_{4}(L) = {α | α form/L, dim α = 2n, α ∈ W_{t}L}

Again, we construct infinite towers of fields F_{1} ⊂ F_{2} ⊂ ⋯ and F_{1} = F′_{1} ⊂ F′_{2} ⊂ · · ·. Suppose we have constructed F_{i} resp. F′_{i}, i ≥ 1. Then we let F_{i+1} (resp. F′_{i+1}) be the compositum of all function fields F_{i}(α) (resp. F′_{i}(α)) where α ∈ C_{1}(F_{i}) ∪ C_{2}(F_{i}) ∪ C_{3}(F_{i}) (resp. C_{1}(F′_{i}) ∪ C_{2}(F′_{i}) ∪ C_{3}(F′_{i}) ∪ C_{4}(F′_{i})).

We then put F = ⋃_{i=1}^{∞} F_{i} (resp. F′ = ⋃_{i=1}^{∞} F′_{i}). Note that since we only take function fields of t.i. forms, all orderings of F_{1} extend to F, resp. F′. In particular, F, F′ will be real.

Now a field F is SAP if and only if all forms of type ⟨1, a, b, −ab⟩ are weakly isotropic, i.e. there exists an n such that the n-fold orthogonal sum n × ⟨1, a, b, −ab⟩ is isotropic (cf. [P, Satz 3.1], [ELP, Th. C]). Thus, taking function fields of forms of type ⟨1, 1⟩ ⊗ ⟨1, x, y, −xy⟩ assures that F (resp. F′) is SAP. Taking function fields of forms in I_{t}^{3} yields that I_{t}^{3}F = 0 (resp. I_{t}^{3}F′ = 0).

We now show that (B_{n})_{K} is a division algebra for K = F, F′. This then implies that λ(K) ≥ n − 1. Let L be an extension of F_{1} such that all orderings of F_{1} extend to L and suppose we have that (B_{n})_{L} is division. Then B_{n} stays division over L(α) for α ∈ C_{j}(L), j = 1, 3, 4, by a reasoning similar to above after invoking Lemma 2.2(iii). Also, B_{n} stays division over K = L(⟨⟨−1, −x, −y⟩⟩) for all x, y ∈ L^{∗} by part (d) of Lemma 2.2(iii). Now α = ⟨1, 1⟩ ⊗ ⟨1, x, y, −xy⟩ contains the Pfister neighbor ⟨1, 1⟩ ⊗ ⟨1, x, y⟩ of ⟨⟨−1, −x, −y⟩⟩, therefore α becomes isotropic over K, hence K(α)/K is purely transcendental and B_{n} stays division over K(α) = L(⟨⟨−1, −x, −y⟩⟩)(α) and therefore over L(α).

This shows that (B_{n})_{K} is a division algebra for K = F, F′. Hence, λ(K) ≥ n − 1. By a similar reasoning, (A_{n})_{F} is a division algebra.

Suppose that λ(K) ≥ n. Then there exists C ∈ Br_{2}(K) such that t(C) = n. Now K is SAP and I_{t}^{3}K = 0. Hence, by Lemma 2.2(i) and Lemma 2.10(iii), there exists an anisotropic Albert form α of dimension 2n + 2 associated with C such that α ≅ α_{0} ⊥ α_{t} with α_{t} ∈ W_{t}F and dim α_{0} ∈ {0, 4}. But such an α is by construction isotropic (consider the forms in C_{1} above !), a contradiction. Hence λ(K) = n − 1. By Theorem 1.4, we get u(K) ∈ {2n − 2, 2n}.

Now over F′, we have by construction that all torsion forms of dimension 2n are isotropic (consider the forms in C_{4} above !). Thus, u(F′) = 2n−2 = 2λ(F′).

We already remarked that (A_{n})_{F} is a division algebra. Hence, its associated Albert form (ψ_{n})_{F} is anisotropic and torsion. Therefore, u(F) ≥ 2n and we necessarily have u(F) = 2n.

It remains to show that ũ(F) = ũ(F′) = ∞. Let m be a positive integer and let μ_{m} = m×⟨1⟩ ⊥ t⟨1, −(1 + x_{1}^{2})⟩ over F_{1}. Since m×⟨1⟩ and ⟨1, −(1 + x_{1}^{2})⟩ are anisotropic over F_{0}(x_{1}, x_{2}, ⋯, y_{1}, y_{2}, ⋯), it follows from Springer’s theorem [L1, Ch. VI, Prop. 1.9] that μ_{m} is anisotropic. Furthermore, μ_{m} is t.i. as ⟨1, −(1 + x_{1}^{2})⟩ is a binary torsion form. Thus, if we can show that μ_{m} stays anisotropic over F (resp. F′) for all m, then ũ(F), ũ(F′) ≥ 2m + 2 for all m and thus ũ(F) = ũ(F′) = ∞.

We now construct a tower of fields L_{1} ⊂ L_{2} ⊂ ⋯ such that L_{i} will be the power series field in the variable t over some L′_{i}, L_{i} = L′_{i}((t)), such that F_{i} ⊂ L_{i} (resp. F′_{i} ⊂ L_{i}), and (μ_{m})_{L_{i}} anisotropic for all m ≥ 0 and all i ≥ 1. This then shows that (μ_{m})_{F_{i}} (resp. (μ_{m})_{F′_{i}}) is anisotropic for all m ≥ 0, i ≥ 1, and therefore (μ_{m})_{F} (resp. (μ_{m})_{F′}) will be anisotropic for all m ≥ 0.

Suppose we have constructed L_{i} = L′_{i}((t)). Note that necessarily L_{i} is real as (μ_{m})_{L_{i}} is anisotropic for all m ≥ 0. Let P_{i} ∈ X_{L′_{i}} be any ordering and M′_{i} be the compositum over L′_{i} of the function fields of all forms (defined over L′_{i}) in

C′(L′_{i}) = {α | α indefinite at P_{i}, dim α ≥ 3} .

Let M_{i} = M′_{i}((t)).

Now let ρ ∈ C_{j}(F_{i}) (resp. C_{j}(F′_{i})), 1 ≤ j ≤ 4. By Springer’s theorem, ρ_{L_{i}} ≅ ρ_{1} ⊥ tρ_{2} with ρ_{k}, k = 1, 2, defined over L′_{i}. We will show that ρ_{k} ∈ C′(L′_{i}) for at least one k ∈ {1, 2}.

First, note that forms in C_{j}(F_{i}) (resp. C_{j}(F′_{i})), 1 ≤ j ≤ 4, are of dimension ≥ 6 (recall that 2 ≤ λ = n − 1). Thus, dim ρ_{k} ≥ 3 for at least one k ∈ {1, 2}. If ρ_{L_{i}} is isotropic, then over L′_{i} we have ⟨1, −1⟩ ⊂ ρ_{k} for at least one k ∈ {1, 2}, and since ⟨1, −1⟩ ≅ t⟨1, −1⟩, we may “shift” the hyperbolic plane from one ρ_{k} to the other if necessary to obtain the desired result, namely that ρ_{k} ∈ C′(L′_{i}) for at least one k ∈ {1, 2}.

Let us therefore assume that ρ_{L_{i}} is anisotropic.

Suppose ρ ∈ C_{1}(F_{i}) (resp. C_{1}(F′_{i})). Then dim ρ ≥ 8, and we can write ρ ≅ η ⊥ τ over F_{i}, with τ torsion and dim τ ≥ 4. Now τ_{L_{i}} ≅ τ_{1} ⊥ tτ_{2} with τ_{k}, k = 1, 2, defined over L′_{i}. Since τ is torsion, we have that τ_{1} and τ_{2} are torsion. Now τ_{k} ⊂ ρ_{k} over L′_{i} by Springer’s theorem as ρ_{L_{i}} is anisotropic, and a simple dimension count shows that there exists at least one k ∈ {1, 2} such that dim τ_{k} ≥ 2 and dim ρ_{k} ≥ 4, which implies that for this k we have ρ_{k} ∈ C′(L′_{i}).

Suppose ρ ≅ ⟨1, 1⟩ ⊗ ⟨1, x, y, −xy⟩ ∈ C_{2}(F_{i}) (resp C_{2}(F′_{i})). Then either ρ_{L_{i}} is already defined over L′_{i}, in which case it is a t.i. form of dimension 8 and thus in C′(L′_{i}). Or there exist a, b ∈ L_{i}^{′∗} such that ρ_{L_{i}} ≅ ⟨1, 1⟩ ⊗ ⟨1, a⟩ ⊥ bt⟨1, 1⟩⊗⟨1, −a⟩. then either ⟨1, 1⟩⊗⟨1, a⟩ is indefinite at P_{i} and thus in C′(L′_{i}), or ⟨1, 1⟩ ⊗ ⟨1, −a⟩ is indefinite at P_{i} and thus in C′(L′_{i}).

Finally, if ρ ∈ C_{j}(F_{i}) (resp. ρ ∈ C_{j}(F′_{i})), j = 3, 4, then ρ is already torsion of dimension ≥ 6 (for j = 3 this follows from the Arason-Pfister Hauptsatz), but then ρ_{1} and ρ_{2} are torsion over L′_{i}, and since at least one of them is necessarily of dimension ≥ 4, the result follows.

Thus, each ρ ∈ C_{j}(F_{i}) (resp. C_{j}(F_{i})), 1 ≤ j ≤ 4 has the property that ρ_{L_{i}} ≅ ρ_{1} ⊥ tρ_{2} with ρ_{k}, k = 1, 2, defined over L′_{i} and ρ_{k} ∈ C′(L′_{i}) for at least one k. But then, (ρ_{k})_{M′_{i}} is isotropic by construction, hence also ρ_{M_{i}}. In particular, M_{i}(ρ)/M_{i} is a purely transcendental extension.

Let us now show that (μ_{m})_{F} is anisotropic for all m. Let N_{i} be the compositum of the function fields of all forms α_{M_{i}} with α ∈ C_{1}(F_{i}) ∪ C_{2}(F_{i}) ∪ C_{3}(F_{i}). By the above, N_{i}/M_{i} is purely transcendental. Let B be a transcendence basis so that N_{i} = M_{i}(B) = M′_{i}((t))(B). We now put L′_{i+1} = M′_{i}(B) and L_{i+1} = L′_{i+1}((t)) = M′_{i}(B)((t)). There are obvious inclusions F_{i+1} ⊂ N_{i} = M′_{i}((t))(B) ⊂ M′_{i}(B)((t)) = L_{i+1}. Since M′_{i} is obtained from L′_{i} by taking function fields of forms indefinite at P_{i}, we see that P_{i} extends to an ordering on M′_{i} and thus clearly also to orderings on L′_{i+1}.

To show that (μ_{m})_{F} is anisotropic, it thus suffices to show that if μ_{m} is anisotropic over L_{i}, then it stays anisotropic over L_{i+1}. Now m × ⟨1⟩ is clearly anisotropic over the real field L′_{i+1}. Also, ⟨1, −(1 + x_{1}^{2})⟩, which is anisotropic over L′_{i} by assumption, stays anisotropic over L′_{i+1} as L′_{i+1} is obtained by taking function fields of forms of of dimension ≥ 3 over L′_{i} followed by a purely transcendental extension. By Springer’s theorem, (μ_{m})_{L_{i+1}} = (m × ⟨1⟩ ⊥ t⟨1, −(1 + x_{1}^{2})⟩)_{L_{i+1}} is anisotropic.

The proof for F′ is the same as above except that we have to take N_{i} above to be the compositum of the function fields of all forms α_{M_{i}} with α ∈ C_{1}(F′_{i}) ∪ C_{2}(F′_{i}) ∪ C_{3}(F′_{i}) ∪ C_{4}(F′_{i}).

The case λ = u = ũ = ∞

This can be done by the same type of construction as above, but this time we only consider function fields of forms of the following types :

\begin{matrix}C_{1}(L) = \{α | α = ⟨1, 1⟩ ⊥ ⟨1, x, y, −xy⟩, x, y ∈ L^{∗}\} \\ C_{2}(L) = \{α | α form/L, α ∈ I_{t}^{3}L\} \end{matrix}

The field F we will obtain has, just as before, the property SAP and I_{t}^{3}F = 0. Furthermore, the algebra A_{n} will stay a division algebra over F for all n ≥ 3. Hence λ(F) = ∞ and it follows immediately that u(F) = ũ(F) = ∞. (Note that for each n ≥ 2 the form (ψ_{n})_{F} will be an anisotropic 2n-dimensional torsion form.)

Now we can prove Corollary 1.6 from the introduction, which we restate in a more detailed version for the reader’s convenience.

Corollary 3.2. Let F be a real field with I_{t}^{3}F = 0. Then

(u(F), ũ(F)) ∈ {(2n, 2n); n ≥ 0} ∪ {(2n, ∞); n ≥ 0} ∪ {(2n, 2n + 2); n ≥ 2} .

All pairs of values on the right hand side can be realized as pairs (u(F), ũ(F)) for suitable real F with I_{t}^{3}F = 0. Furthermore, there exist such F which are SAP with the only exceptions being the pairs (0, ∞), (2, ∞).

Proof. Let us first show that no other values are possible. By Lemma 2.4, u and ũ are always even or infinite. If F in non-SAP, then ũ(F) = ∞. So suppose that F is SAP. If u(F) ≤ 2, then ũ(F) ≤ 2 by [ELP, Theorems E,F], and it follows readily that u(F) = ũ(F) ∈ {0, 2}. Note that this also shows that (0, ∞), (2, ∞) cannot be realized by SAP-fields. If F is linked, then by Theorem 1.1, u(F) = ũ(F) ∈ {0, 2, 4, 8}. If, however F is non-linked, then Theorem 1.5 (3.1) shows that there can be no other pairs (u, ũ) than the ones in the statement of the corollary.

The pairs (u, ũ) = (0, 0) (resp. (2, 2)) can be realized by ℝ (resp. the rational function field in one variable over the reals, ℝ(X)). Real global fields have (u, ũ) = (4, 4). (u, ũ) = (0, ∞) is realized by ℝ((X))((Y)), see also Example 2.12. Examples of fields with (u, ũ) = (2, ∞) can be found in [EP, Cor. 5.2]. All other combinations have been realized in Theorem 1.5 (3.1) by SAP-fields.

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Detlev W. Hoffmann Laboratoire de Mathématiques UMR 6623 du CNRS Université de Franche-Comté 16 route de Gray F-25030 Besançon Cedex detlev@math.univ-fcomte.fr