On the Number of Square Classes

of a Field of Finite Level

Karim Johannes Becher

Received: May 29, 2001 Revised: October 31, 2001

Communicated by Ulf Rehmann

Abstract. The level question is, whether there exists a field F with finite square class number q(F) := |F^{×}/F^{×}^{2}| and finite level s(F) greater than four. While an answer to this question is still not known, one may ask for lower bounds for q(F) when the level is given.

For a nonreal field F of level s(F) = 2^{n}, we consider the filtration of the groups D_{F} (2^{i}), 0 ≤ i ≤ n, consisting of all the nonzero sums of 2^{i} squares in F. Developing further ideas of A. Pfister, P. L. Chang and D. Z. Djoković and by the use of combinatorics, we obtain lower bounds for the invariants \overline{q}_{i} := |D_{F} (2^{i})/D_{F} (2^{i−1})|, for 1 ≤ i ≤ n, in terms of s(F). As a consequence, a field with finite level ≥ 8 will have at least 512 square classes. Further we give lower bounds on the cardinalities of the Witt ring and of the 2-torsion part of the Brauer group of such a field.

1 Introduction

Let F be a field. The level of F, denoted by s(F), is defined as the least positive integer m such that −1 is a sum of m squares in F whenever such an integer exists and ∞ otherwise. For fields of positive characteristic this invariant can take only the values 1 and 2, depending just on whether −1 is a square in F or not. Fields of level ∞, i.e. in which −1 is not a sum of squares, are called real fields and an equivalent condition to s(F) = ∞ is the existence of an ordering on F. Fields of finite level are also called nonreal fields. For a long time it has been an open question which values exactly occur as the level of some field. The complete solution to this problem was given by A. Pfister in [10] and it inspired a big part of later advances in the theory of quadratic forms, e.g. the development of the theory of Pfister forms and the investigation of isotropy behaviors of quadratic forms under function field extensions.

Pfister proved that the level of a nonreal field is always a power of 2 [10, Satz 4] and further that, if F is any real field (e.g. ℚ or ℝ) and n ≥ 0, then the function field of the projective quadric X_{0}^{2} + ⋯ + X_{2^{n}}^{2} = 0 over F has level 2^{n} [10, Satz 5]. These were the first examples of nonreal fields of level greater than 4 and, actually, still no examples of an essentially different kind are known. In general it remains a difficult problem to determine the level of a given field of characteristic zero. For an overview on what is known about levels of common types of fields we refer to [8, Chap. XI, Section 2]. In the same book T. Y. Lam also mentions the following question [8, p. 333]:

1.1. Level Question. Does there exist a field F such that 4 < s(F) < ∞ and such that F^{×}/F^{×}^{2} is finite?

Here and in the sequel we denote by F^{×} the multiplicative group of F and by F^{×}^{2} the subgroup of nonzero squares in F. The quotient F^{×}/F^{×}^{2} is called the square class group of F. We call q(F) := |F^{×}/F^{×}^{2}| the square class number of F. Another subgroup of F^{×} of importance is the group of nonzero sums of squares in F, denoted as ∑F^{×}^{2}. Further, for any m ∈ IN we denote by D_{F} (m) the set of elements of F^{×} which can be written as a sum of m squares over F. Pfister has shown that D_{F} (m) is a group whenever m is a power of 2 [10, Satz 9]. We thus have the following group filtration for ∑F^{×}^{2}:

F^{×}^{2} ⊊ D_{F} (2) ⊊ D_{F} (4) ⊊ ⋯ ⊊ D_{F} (2^{i−1}) ⊊ D_{F} (2^{i}) ⊊ ⋯ ⊂ ∑F^{×}^{2}. (1.2)

If F is nonreal of level 2^{n} then we actually have D_{F} (2^{n} + 1) = ∑F^{×}^{2} = F^{×}. For i ≥ 1 we define q̄_{i}(F) := |D_{F} (2^{i})/D_{F} (2^{i−1})|. Note that the quotients F^{×}/F^{×}^{2} and D_{F} (2^{i})/D_{F} (2^{i−1}) are 2-elementary abelian groups. So q(F) and q̄_{i}(F) are each either a power of 2 or ∞. From (1.2) we see that the inequality

q(F) ≥ q̄_{1}(F) ⋯ q̄_{n}(F)  (1.3)

holds for any n ≥ 1. We will use this in particular when s(F) = 2^{n}.

While an answer to the level question is still not known, one may look for lower bounds on |F^{×}/F^{×}^{2}| in terms of s(F). One approach is to search for lower bounds on the invariants q̄_{i}(F) and to use then (1.3) to obtain a bound for q(F). Following this idea, A. Pfister obtained in [11, Satz 18.d] the following estimate for a field F of level 2^{n}:

q(F) ≥ 2^{\frac{n(n+1)}{2}} .  (1.4)

His proof (see also [8, p. 325]) actually shows for 1≤i≤n that

q̄_{i}(F)≥2^{n+1−i}.  (1.5)

On the Number of Square Classes … 67

Our standard examples of fields of level 1, 2 and 4, respectively, are the field of complex numbers ℂ, the finite field 𝔽_{3} and ℚ_{2}, the field of dyadic numbers. These examples show that (1.4) is best possible for n ≤ 2. For higher n, however, P. L. Chang has improved the bound using combinatorics. In [1] he shows that q(F) ≥ 128 for a field F of level eight and further that q(F) ≥ 16· \frac{2^{s}}{s^{2}} for any nonreal field F of level s ≥ 16. His approach has been refined by D. Ž. Djoković in [2], leading to the following estimate:

q(F) ≥ 2 · ∑_{i=1}^{s/2} \frac{1}{s + 2 − i} ( \begin{matrix}s + 1 \\ i \end{matrix} ) > \frac{2^{s}}{s} .  (1.6)

Their method does not provide any information about the invariants q̄_{i}(F). The aim of the present work is to extend this method and to get lower bounds for the invariants q̄_{i}(F) with respect to s(F) which improve (1.5). The combinatorial aspect is postponed to the two appendices where a certain coloring problem for (hyper-)graphs is considered. We use common notations and results from quadratic form theory; the standard references are [8] and [12]. (Note that the uncomfortable case of characteristic 2 is implicitly excluded whenever we deal with a field of level greater than 1.) For isometry of quadratic forms we use the symbol ≅ . For a quadratic form φ over F we denote by D_{F} (φ) the set of nonzero elements of F represented by φ. We sometimes say just “form” or “quadratic form” to mean “non-degenerate quadratic form”. A diagonalized quadratic form over F with coefficients a_{1}, …, a_{m} ∈ F^{×} is denoted by ⟨a_{1}, …, a_{m}⟩. An m-fold Pfister form is a quadratic form of the shape ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{m}⟩ and shortly written as ⟨⟨a_{1}, …, a_{m}⟩⟩; its dimension is 2^{m}. A neighbor of an m-fold Pfister form π is a quadratic form φ which is similar to a subform of π and of dimension greater than 2^{n−1}. We know that in this situation φ is isotropic if and only of π is hyperbolic. By W (F) we denote the Witt ring of F, further by Br(F) the Brauer group and by Br_{2}(F) its 2-torsion part. In (3.1), (5.4) and (5.5) we shall use Milnor Ktheory. For definitions and properties of the Milnor ring k_{∗}F and its homgenous components k_{m}F (m ≥ 0) we refer to [9] and [3]. However, we use the notation {a_{1}, …, a_{m}} instead of ℓ(a_{1}) ⋯ ℓ(a_{m}) for a symbol in k_{m}F. We recall that this symbol is zero in k_{m}F if and only if the corresponding m-fold Pfister form ⟨⟨−a_{1}, …, −a_{m}⟩⟩ over F is hyperbolic (see [3, Main Theorem 3.2]). In particular, s(F) = 2^{n} is equivalent to {−1}^{n} ≠ 0 and {−1}^{n+1} = 0 in k_{∗}F. Everywhere else in the text, {x_{1}, …, x_{n}} stands simply for the set of elements x_{1}, …, x_{1}.

Acknowledgments

This work is part of the author’s Ph.D. thesis research, carried out at the Université de Franche-Comté under the supervision of Eva Bayer-Fluckiger and Detlev W. Hoffmann. The author expresses his gratitude to his supervisors as well as to Ján Mináč and to David B. Leep for encouragement and stimulating discussions on the subject.

2 Sums of squares in fields

Let F be a field. For an element x ∈ F we define its length (over F) to be the least positive integer m such that x can be written as a sum of m nonzero squares over F if such an integer exists and ∞ otherwise (i.e. if x is not a nontrivial sum of nonzero squares over F). We denote this value in IN ∪ {∞} by ℓ_{F} (x), or just by ℓ(x) whenever the context makes clear over which field F we are working. Obviously ℓ_{F} (x) depends on x only up to multiplication by a nonzero square in F; in other words, ℓ_{F} (x) is an invariant of the square class xF^{×}^{2} whenever x ≠ 0. For m ≥ 1, D_{F} (m) is by definition the set {x ∈ F^{×} | ℓ(x) ≤ m}. Our investigation into lengths of field elements is based on the following famous result [10, Satz 2]:

2.1. Theorem (Pfister). For any i ≥ 0, D_{F} (2^{i}) is a subgroup of F^{×}.

A simple proof within the theory of Pfister forms can be found in [12, 4.4.1. Lemma]. As a consequence of this theorem one gets an inequality linking the lengths of two elements to the length of their product. We include a proof of this result, which is [10, Satz 3].

2.2. Lemma. For any x, y ∈ F we have the inequalities ℓ(x + y) ≤ ℓ(x) + ℓ(y) and ℓ(xy) ≤ ℓ(x) + ℓ(y) − 1.

Proof: The first inequality is obvious from the definition of the length. The second inequality is trivial if xy is zero or if x or y is not a sum of squares. So we may suppose that both x and y are nonzero sums of squares in F. Let then r be the least nonnegative integer such that x, y ∈ D_{F} (2^{r}). We will prove ℓ(xy) < ℓ(x) + ℓ(y) by induction on r. If r = 0 then x, y and xy are squares in F and the inequality is clear. Suppose now that r > 0. Since D_{F} (2^{r}) is a group we know that ℓ(xy) ≤ 2^{r}. So the inequality is clear if 2^{r} < ℓ(x) + ℓ(y). Otherwise, we may suppose that ℓ(y) ≤ 2^{r−1}. By the choice of r we then have 2^{r−1} < ℓ(x) ≤ 2^{r} and may therefore write x = a + z with a, z ∈ F^{×} such that ℓ(a) = 2^{r−1} and ℓ(z) = ℓ(x) − 2^{r−1} ≤ 2^{r−1}. By the induction hypothesis we have ℓ(zy) < ℓ(y) + ℓ(z). As D_{F} (2^{r−1}) is a group we have ℓ(ay) ≤ 2^{r−1}. Since xy = ay + zy, using the first inequality of the statement we obtain finally ℓ(xy) ≤ ℓ(ay) + ℓ(zy) < 2^{r−1} + ℓ(y) + ℓ(z) = ℓ(x) + ℓ(y). □

According to the definition we gave in the introduction, the level of F is the length of −1 in F. We may also conclude that ℓ_{F} (0) = s(F) + 1. Therefore, from any of the inequalities of the lemma we obtain immediately:

2.3. Corollary. For any x ∈ F we have ℓ(x) + ℓ(−x) ≥ s(F) + 1. □

2.4. Corollary. Let a_{1}, …, a_{m} ∈F^{×}. If the quadratic form ⟨a_{1}, …, a_{m}⟩ over F represents the element x ∈ F nontrivially then ℓ(a_{1}) + ⋯ + ℓ(a_{m}) ≥ ℓ(x).

Proof: If the form ⟨a_{1}, …, a_{m}⟩ represents x ∈ F nontrivially, this means that there are x_{1}, …, x_{m} ∈ F, not all zero, such that a_{1}x_{1}^{2} + ⋯ + a_{m}x_{m}^{2} = x. We may suppose that x_{i} is nonzero for 1 ≤ i ≤ m′ and zero for m′ < i ≤ m. From the first inequality of the lemma we obtain ℓ(x) ≤ ℓ(a_{1}x_{1}^{2}) + ⋯ + ℓ(a_{m′}x_{m′}^{2}) = ℓ(a_{1}) + ⋯ + ℓ(a_{m′}). □

For i ≥ 0, we say that the elements a_{1}, …, a_{m} ∈ F^{×} are independent modulo D_{F} (2^{i}) if in F^{×}/D_{F} (2^{i}), considered as an 𝔽_{2}-vectorspace, the classes represented by a_{1}, …, a_{m} are 𝔽_{2}-linear independent.

2.5. Proposition. For i ≥ 2, let a, b ∈ D_{F} (3·2^{i−2})\D_{F} (2^{i−1}) and c ∈ D_{F} (2^{i}) such that ℓ(a + b + c) > 2^{i+1}. Then the elements a, b and c of D_{F} (2^{i}) are independent modulo D_{F} (2^{i−1}).

Proof: We have to show that a, b, c, ab, ac, bc, abc ∉ D_{F} (2^{i−1}). For a and b this is already given. We put x := a + b + c. Each of the quadratic forms ⟨a, b, c⟩, ⟨a, b, abc⟩, ⟨1, ab, ac⟩ and ⟨ac, bc, 1⟩ over F represents one of the elements x, abx, ax and cx and neither of these elements lies in the group D_{F} (2^{i+1}). We obtain from (2.4) that each of the numbers ℓ(a) + ℓ(b) + ℓ(c), ℓ(a) + ℓ(b) + ℓ(abc), 1 + ℓ(ab) + ℓ(ac) and ℓ(ac) + ℓ(bc) + 1 is greater than 2^{i+1}. Since ℓ(a) + ℓ(b) ≤ 3 · 2^{i−1} and ab, ac, bc ∈ D_{F} (2^{i}) we obtain ℓ(c), ℓ(abc) ≥ 2^{i−1} and further ℓ(ab) = ℓ(ac) = ℓ(bc) = 2^{i}. □

For the rest of this section we fix a sum of squares

x = x_{1}^{2} + ⋯ + x_{l}^{2}  (2.6)

with x_{1}, …, x_{l} ∈ F^{×}, x ∈ F and l = ℓ_{F} (x). For a subset I ⊂ {1, …, l} we denote x_{I} := ∑_{i∈I} x_{i}^{2}. If I is not empty then we have ℓ(x_{I}) = |I|. For a real number z we denote by ⌈z⌉ the least integer ≥ z.

2.7. Theorem. Let I and J be nonempty proper subsets of {1, …, l}. Let r be a nonnegative integer such that x_{I}x_{J} ∈ D_{F} (2^{r}). Then the following hold:

(i) ⌈\frac{|I|}{2^{r}}⌉ = ⌈\frac{|J|}{2^{r}}⌉, in particular | |I| − |J| | < 2^{r},

(ii) |I \ J| , |J \ I| ≤ 2 ℓ(x_{I}x_{J}) − 1 < 2^{r+1},

(iii) |I ∪ J| − |I ∩ J| ≤ 2^{r+1} + ℓ(x_{I}x_{J}) − 1 ≤ 3 · 2^{r} − 1.

Proof: The hypothesis implies that x_{I} and x_{J} are nonzero elements of F. We set m := ℓ(x_{I}x_{J}) and a := \frac{x_{J}}{x_{I}}. Then ℓ(a) = m ≤ 2^{r}. If ν is an integer such that |I| ≤ ν2^{r} then we can write x_{I} as a sum of ≤ ν elements of D_{F} (2^{r}). As D_{F} (2^{r}) is a group, x_{J} = ax_{I} can also be written as a sum of ≤ ν elements of D_{F} (2^{r}) which means that |J| = ℓ(x_{J}) ≤ ν2^{r}. By symmetry we obtain for any ν ∈ IN that |I| ≤ ν2^{r} if and only if |J| ≤ ν2^{r}. This shows (i).

We compute x_{I∪J} = x_{I\J} + x_{J} = (1 + a)x_{I\J} + ax_{I∩J} and then substitute y := (1+a)x_{I\J} and z := ax_{I∩J} to have x_{I∪J} = y + z. If y ≠ 0 then we have ℓ(y) ≤ m + |I \ J| by (2.2), but also ℓ(y) ≤ 2^{r+1} since D_{F} (2^{r+1}) is a group. If z ≠ 0 then (2.2) yields ℓ(z) ≤ m + |I ∩ J| − 1. Therefore, if at least one of y and z is nonzero then we obtain the inequalities ℓ(y + z) ≤ |I| + 2m − 1 and ℓ(y + z) ≤ 2^{r+1} + m + |I ∩ J| − 1. Both inequalities remain valid in the case y = z = 0, since then necessarily a = −1, whence ℓ(y + z) = ℓ(0) = m + 1. As |I ∪ J| = ℓ(y + z) we obtain (ii) by symmetry from the first and (iii) from the second inequality. □

For m = 1 this leads to an observation made in the proof of [1, Theorem 1]:

2.8. Corollary (Chang). Let I and J be as in the theorem. If x_{I} and x_{J} lie in the same square class then both sets have the same cardinality and differ by at most one element. □

2.9. Corollary. Let I and J be as in the theorem with |I| = |J| = 2^{i}, i ≥ 2. If x_{I} and x_{J} represent the same class modulo D_{F} (2^{i−1}) then |I ∩J| ≥ 2^{i−2} +1.

Proof: If x_{I} and x_{J} lie in the same class modulo D_{F} (2^{i−1}) then ℓ(x_{I}x_{J})≤ 2^{i−1}. Applying part (iii) of the theorem for r = i − 1 we obtain |I ∪ J| − |I ∩ J| ≤ 3·2^{i−1} −1. But our hypothesis here gives |I ∪J| = 2·2^{i} −|I ∩J|. This together implies |I ∩ J| > 2^{i−2}. □

3 The invariants q̄_{i}

For a nonreal field F of level 2^{n} we are going to study the invariants q̄_{i}(F) = |D_{F} (2^{i})/D_{F} (2^{i−1})| for 1 ≤ i ≤ n. In particular, we are interested to know whether Pfister’s bounds (1.5) can be improved. First we note that the bound q̄_{n}(F) ≥ 2, obtained from (1.5) for i = n, just takes into account that −1 represents a nontrivial class in the group D_{F} (2^{n})/D_{F} (2^{n−1}). In spite of the simple argument, this bound is optimal for every n ≥ 1. More precisely, for any n ≥ 1 there is a field F of level 2^{n} such that F^{×} = D_{F} (2^{n−1}) ∪ −D_{F} (2^{n−1}). The construction of such an example will be included in a forthcoming paper of the author. We now turn to consider q̄_{n−1}(F). For i = n − 1, (1.5) gives q̄_{n−1}(F) ≥ 4. The example F = ℚ_{2} shows that this bound is optimal for n = 2.

3.1. Theorem. Let F be a field of level 2^{n} with n ≥ 3. Then q̄_{n−1}(F) ≥ 16.

Proof: Since ℓ(0) = 2^{n} + 1 and n ≥ 3, we may choose elements a_{1}, a_{2}, a_{3} ∈ F^{×} such that a_{1} + a_{2} + a_{3} = 0 and 2^{n−2} + 1 ≤ ℓ(a_{i}) ≤ 3 · 2^{n−3} for i = 1, 2, 3. Then by (2.5), a_{1}, a_{2} and a_{3} are independent modulo D_{F} (2^{n−2}). Let H be the subgroup of D_{F} (2^{n−1}) generated by D_{F} (2^{n−2}) and the elements a_{1}, a_{2} and a_{3}. Since |H/D_{F} (2^{n−2})| = 8 it remains to show that H ≠ D_{F} (2^{n−1}).

To this aim, we will calculate in the Milnor ring k_{∗}F. For i = 1, 2, 3 we fix the symbols β_{i} := {a_{1}a_{2}a_{3}, a_{i}} and γ_{i} := {−a_{1}a_{2}a_{3}, −a_{i}} in k_{2}F. Let ε denote the element {−1} in k_{1}F. Since s(F) = 2^{n} we have ε^{n} ≠ 0. As a_{1}, a_{2}, a_{3} ∈ D_{F} (2^{n−1}) we observe that β_{1}+β_{2}+β_{3} = {−1, a_{1}a_{2}a_{3}} is annihilated by ε^{n−2} and that ε^{n−2}(β_{i}+γ_{i}) = ε^{n−2}({a_{1}a_{2}a_{3}, −1}+{−1, a_{i}}+{−1, −1}) = ε^{n} for i = 1, 2, 3. If ε^{n−2}β_{i} ≠ 0 in k_{n}F for some i then by the above relations we may suppose that ε^{n−2}β_{i} ≠ 0 for i = 1, 2 and ε^{n−2}β_{3} ≠ ε^{n}, i.e. ε^{n−2}γ_{3} ≠ 0. Using that a_{1} + a_{2} + a_{3} = 0 we compute {−a_{2}, −a_{3}} = {a_{1}, −a_{2}a_{3}} = β_{1} and equally {−a_{1}, −a_{3}} = β_{2}. Since none of β_{1}, β_{2} and γ_{3} is annihilated by ε^{n−2}, the symbols ε^{n−2}{−a_{2}, −a_{3}}, ε^{n−2}{−a_{1}, −a_{3}} and ε^{n−2}{−a_{1}a_{2}, −a_{3}} in k_{n}F are all nonzero. Therefore the Pfister forms 2^{n−2}×⟨⟨a_{2}, a_{3}⟩⟩, 2^{n−2} ×⟨⟨a_{1}, a_{3}⟩⟩ and 2^{n−2}×⟨⟨a_{1}a_{2}, a_{3}⟩⟩ are anisotropic. Further, 2^{n−2}×⟨⟨1, a_{3}⟩⟩ ≅ 2^{n}×⟨1⟩ is anisotropic since s(F) = 2^{n}. This shows that −1, −a_{1}, −a_{2}, −a_{1}a_{2} ∉ D_{F} (2^{n−2}×⟨⟨a_{3}⟩⟩). As the group D_{F} (2^{n−2}×⟨⟨a_{3}⟩⟩) contains the subgroup D_{F} (2^{n−2}) and the element a_{3} we conclude that D_{F} (2^{n−2} ×⟨⟨a_{3}⟩⟩) ∩ −H = ∅. On the other hand, since ℓ(−a_{3}) ≤ ℓ(a_{1})+ℓ(a_{2}) ≤ 3·2^{n−2} we can write −a_{3} = x+y with x ∈ D_{F} (2^{n−1}), y ∈ D_{F} (2^{n−2}) and obtain −x = y + a_{3} ∈ D_{F} (2^{n−2}×⟨⟨a_{3}⟩⟩) ∩ −D_{F} (2^{n−1}). Now we study the case where ε^{n−2}β_{i} = 0 for i = 1, 2, 3. As ε^{n−2}β_{i} = ε^{n−2}{−a_{1}a_{2}a_{3}, a_{i}}, this means that the Pfister form 2^{n−2} × ⟨⟨a_{1}a_{2}a_{3}, −a_{i}⟩⟩ is hyperbolic for i = 1, 2, 3. We conclude that H ⊂ D_{F} (2^{n−2} × ⟨⟨a_{1}a_{2}a_{3}⟩⟩). As the Pfister form 2^{n−1} × ⟨⟨a_{1}a_{2}a_{3}⟩⟩ ≅ 2^{n} × ⟨1⟩ is anisotropic we have −1 ∉ D_{F} (2^{n−2} × ⟨⟨a_{1}a_{2}a_{3}⟩⟩) and therefore D_{F} (2^{n−2} × ⟨⟨a_{1}a_{2}a_{3}⟩⟩) ∩ −H = ∅. Since −a_{1}a_{2}a_{3} = a_{1}^{2}a_{2} + a_{2}^{2}a_{1} we have ℓ(−a_{1}a_{2}a_{3}) ≤ ℓ(a_{2}) + ℓ(a_{1}) ≤ 3 · 2^{n−2} and may therefore write −a_{1}a_{2}a_{3} = x+y with x ∈ D_{F} (2^{n−1}) and y ∈ D_{F} (2^{n−2}) to obtain this time −x = y + a_{1}a_{2}a_{3} ∈ D_{F} (2^{n−2} × ⟨⟨a_{1}a_{2}a_{3}⟩⟩) ∩ −D_{F} (2^{n−1}). In both cases we have found an element x ∈ D_{F} (2^{n−1}) \ H. □

While the lower bound on q̄_{n−1} of the last theorem is based upon several algebraic arguments, the improvement (with respect to (1.5)) for the lower bounds on q̄_{i}(F) for 2 ≤ i ≤ n − 2 which we present now, is obtained by combinatorial reasoning, developed in appendix A. For integers 0 ≤ k ≤ l we denote by P_{k}^{l} the set of subsets of {1, …, l} with exactly k elements.

3.2. Theorem. Let F be a field of level 2^{n}. Then

q̄_{i}(F) ≥ \{ \begin{matrix}2^{7} & for i = n−2 ≥ 3 , \\ 2^{(n−i)(2^{n−i}+1)+1} & for \frac{n+1}{2} < i ≤ n−3 , \\ 2^{(n−i)(2^{i−2}+1)+1} & for 2 ≤ i ≤ \frac{n+1}{2} . \end{matrix}

Proof: We fix elements x_{1}, …, x_{2^{n}} ∈ F^{×} such that x_{1}^{2} + ⋯ + x_{2^{n}}^{2} = −1. For a subset J ⊂ {1, …, 2^{n}} we denote x_{J} := ∑_{j∈J} x_{j}^{2}. Let 2 ≤ i ≤ \frac{n+1}{2}. We consider the map f : P_{2^{i}}^{2^{n}} ⟶ D_{F} (2^{i})/D_{F} (2^{i−1}) which sends a 2^{i}-subset J ⊂ {1, …, 2^{n}} to the class x_{J}D_{F} (2^{i−1}). By (2.9), if J_{1}, J_{2} ∈ P_{2^{i}}^{2^{n}} are such that f(J_{1}) = f(J_{2}) then |J_{1} ∩ J_{2}| ≥ 2^{i−2} + 1. Therefore (A.8) in appendix A shows |D_{F} (2^{i})/D_{F} (2^{i−1})| ≥ |Im(f)| > 2^{r} for r := (n − i)(2^{i−2}+1). Since D_{F} (2^{i})/D_{F} (2^{i−1}) is a 2-elementary abelian group it must then have at least 2^{r+1} elements. This establishes the third case in the statement. In the remaining cases we cannot apply (A.8) directly for i and m := n. In the case \frac{n+1}{2} <i≤n−3 we have n ≥ 8 and i ≥ 5 and define n′ := 2(n − i + 1) and i′ := n − i + 2 = \frac{n′}{2} + 1. In the case i = n − 2 and n ≥ 5 we set instead n′ := 5 and i′ := 3 = \frac{n′+1}{2}. Note that in both cases n′ − i′ = n − i. For 1 ≤ ν ≤ 2^{n′} let J_{ν} := {(ν − 1) · 2^{n−n′}+ 1, …, ν · 2^{n−n′}} and y_{ν} := x_{J_{ν}}. This yields y_{1} + ⋯ + y_{2^{n′}} = −1 and ℓ(y_{ν}) = |J_{ν}| = 2^{n−n′} for 1 ≤ ν ≤ 2^{n′}. Now we consider the map f′ : P_{2^{i′}}^{2^{n′}} ⟶ D_{F} (2^{i})/D_{F} (2^{i−1}) which sends a 2^{i′}-subset N ⊂ {1, …, 2^{n′}} to the class (∑_{ν∈N}y_{ν})D_{F} (2^{i−1}).

Suppose that f′(N_{1}) = f′(N_{2}) for N_{1}, N_{2} ∈ P_{2^{i′}}^{2^{n′}}. For k = 1, 2 let I_{k} := ⋃_{ν∈N_{k}}J_{ν} ∈ P_{2^{i}}^{2^{n}}. Since by hypothesis ∑_{ν∈N_{1}}y_{ν} = x_{I_{1}} and ∑_{ν∈N_{2}}y_{ν} = x_{I_{2}} lie in the same class of D_{F} (2^{i})/D_{F} (2^{i−1}), (2.9) shows that |I_{1} ∩ I_{2}| ≥ 2^{i−2} + 1 and it follows that |N_{1} ∩ N_{2}| ≥ 2^{i−2−(n−n′)} + 1 = 2^{i′−2} + 1. Having established this intersection property of f′, we obtain from (A.8) that |D_{F} (2^{i})/D_{F} (2^{i−1})| ≥ |Im(f′)| > 2^{r′} holds for r′ := (n′ − i′)(2^{i′−2} + 1). As before, we conclude that |D_{F} (2^{i})/D_{F} (2^{i−1})| ≥ 2^{r′+1}. This finishes the proof since r′ = 6 in case i = n − 2 and r′ = (n − i)(2^{n−i} + 1) otherwise. □

4 Nonreal fields with q̄_{1} equal to the level

From (1.5) we know that q̄_{1}(F) ≥ s(F) holds for any nonreal field F. This bound is optimal for fields of level 1, 2 and 4 as the standard examples show (see introduction). For nonreal fields of higher level, however, there is still no known example where q̄_{1}(F) < ∞. We show that q̄_{1}(F) = s(F) < ∞ is a rather strong condition, with several consequences on the quadratic form structure of F. In particular, for s(F) ≥ 8 it implies that q̄_{2}(F) ≥ \frac{s(F)^{2}}{2} (4.9).

Let ξ be an element of length l ≥ 3 of F. We fix a representation of ξ as a sum of l squares

ξ = x_{1}^{2} + ⋯ + x_{l}^{2}  (4.1)

with x_{1}, …, x_{l} ∈ F^{×}. Let f : P_{2}^{l} → D_{F} (2)/F^{×}^{2} be the function which sends a (nonordered) pair of distinct i, j ≤ l to the square class of x_{i}^{2} + x_{j}^{2}. Considering the elements of D_{F} (2)/F^{×}^{2} as a set of colors, we can interprete f as an edgecoloring of a complete graph in l vertices v_{1}, …, v_{l}. We denote this graph together with its edge-coloring f by G. If in this graph two edges [v_{i}, v_{j}] and [v_{i′}, v_{j′}] are of the same color (with {i, j}, {i′, j′} ∈ P_{2}^{l}) this means that x_{i}^{2} + x_{j}^{2} and x_{i′}^{2} + x_{j′}^{2} lie in the same square class of F, which by (2.8) implies that the sets {i, j} and {i′, j′} intersect. In other words, two edges of the same color in G need to have a vertex in common, i.e. G is a CC-graph in the terminology of appendix B. We get from (B.1) that at least l − 2 colors appear in G. Furthermore, since x_{1}^{2} + ⋯ + x_{l}^{2} is of length l, no sum x_{i}^{2} + x_{j}^{2} with i ≠ j can be a square. This gives a proof of [13, Theorem 1]:

4.2. Proposition (Tort). In (4.1), the partial sums x_{i}^{2} +x_{j}^{2} with 1≤i<j ≤l represent at least l − 2 different nontrivial classes of D_{F} (2)/F^{×}^{2}. □

Let now F be a nonreal field of level s = 2^{n}. We then can choose ξ := 0, which is of length s + 1 over F, and write (4.1) as

0 = x_{1}^{2} + ⋯ + x_{s+1}^{2} .  (4.3)

By the above proposition the partial sums x_{i}^{2} + x_{j}^{2} (with 1 ≤ i < j ≤ s + 1) represent at least s − 1 nontrivial classes of D_{F} (2)/F^{×}^{2}. This shows:

4.4. Corollary. Let F be a nonreal field of level s. Then q̄_{1}(F) ≥ s. Moreover, if q̄_{1}(F) = s then, given any representation (4.3) of zero as a sum of s+1 nonzero squares over F, every nontrivial class of D_{F} (2)/F^{×}^{2} is represented by a partial sum x_{i}^{2} + x_{j}^{2} with 1≤i<j ≤s+1. □

Given a subgroup G ⊂ F^{×}/F^{×}^{2} of finite order 2^{m} we may choose an irredundant set of representatives a_{1}, …, a_{2^{m}} ⊂ F^{×} of the square classes in G and define the quadratic form π_{G} := ⟨a_{1}, …, a_{2^{m}}⟩. Up to isometry, this form does only depend on G and not on the particular choice of the a_{i}. If we choose the a_{i} such that a_{1}, … a_{m} are independent modulo F^{×}^{2} then π_{G} is equal to ⟨⟨a_{1}, …, a_{m}⟩⟩, hence π_{G} is an m-fold Pfister form. If q̄_{1}(F) is finite we write π_{D(2)} for π_{G} with G := D_{F} (2)/F^{×}^{2}.

4.5. Proposition. Let F be a nonreal field with s(F) > 1 and q̄_{1}(F) < ∞. Then π_{D(2)} is hyperbolic.

Proof: Let s := s(F). Given a representation (4.3) of zero as sum of s + 1 squares over F we define a_{i} := x_{2i−1}^{2} + x_{2i}^{2} for 1 ≤ i ≤ s/2. By (2.8) the a_{i} lie in distinct nontrivial square classes. Since a_{1} + ⋯ + a_{s/2} + x_{s+1}^{2} = 0 the form ⟨1, a_{1}, …, a_{s/2}⟩ is isotropic. On the other hand, this is a subform of the Pfister form π_{D(2)}, which then must be hyperbolic. □

4.6. Lemma. Let H be a subgroup of F^{×} containing F^{×}^{2} such that H/F^{×}^{2} is of order 2^{m} with m ≥ 2. If a, b, c, d ∈ H, lie in distinct square classes then there are a_{3}, …, a_{m} ∈ H such that π_{H} = ⟨a, b, c, d⟩ ⊗ ⟨⟨a_{3}, …, a_{m}⟩⟩.

Proof: It is easy to verify that, given four distinct elements t, u, v, w in a 2-elementary abelian group G there exists a subgroup K of index 4 in G such that t, u, v, w represent the four classes of G/K. We apply this fact to the square classes aF^{×}^{2}, bF^{×}^{2}, cF^{×}^{2} and dF^{×}^{2} in G := H/F^{×}^{2}. A subgroup K with the stated property must have order 2^{m−2}. We choose elements a_{3}, …, a_{m} ∈ F^{×} such that their square classes form an 𝔽_{2}-basis of K. The rest is clear. □

4.7. Proposition. Let F be a field with q̄_{1}(F) = s(F) = 2^{n}, n ≥ 2, and let a, b, c, d be elements of D_{F} (2) which lie in distinct square classes.

(a) If a ∉ F^{×}^{2} then D_{F} (⟨1, 1⟩) ∩ D_{F} (⟨1, a⟩) = {1, a}F^{×}^{2}.

(b) If x ∈ D_{F} (⟨1, a⟩) ∩ D_{F} (⟨1, b⟩) ∩ D_{F} (⟨1, c⟩) then ℓ(−x) = 2^{n}.

(c) D_{F} (⟨a, b⟩) ∩ D_{F} (⟨c, d⟩) = ∅.

(d) If n ≥ 3 then D_{F} (⟨a, b⟩) ∩ D_{F} (⟨a, c⟩) ∩ D_{F} (⟨b, c⟩) = ∅.

(e) If x ∈ D_{F} (⟨1, a⟩) ∩ D_{F} (⟨1, b⟩) then ℓ(cx) = 4 or ℓ(−x) ≥ 2^{n} − 1.

Proof: (a) Given a and b lying in distinct nontrivial classes of D_{F} (2)/F^{×}^{2} we may choose a_{3}, …, a_{2^{n−1}} ∈ D_{F} (2) such that φ := ⟨1, a, b, a_{3} …, a_{2^{n−1}}⟩ is a neighbor of the Pfister form π_{D(2)} which is hyperbolic by the last proposition. So φ is isotropic. Now b ∈ D_{F} (⟨1, a⟩) would imply that φ is isometric to ⟨1, 1, ab, a_{3} …, a_{2^{n−1}}⟩ which is a subform of 2^{n} × ⟨1⟩. This is impossible since the latter form is anisotropic by the hypothesis that s(F) = 2^{n}.

(b) Let x ∈ D_{F} (⟨1, a⟩) ∩ D_{F} (⟨1, b⟩) ∩ D_{F} (⟨1, c⟩) where a, b, c ∈ D_{F} (2) are distinct modulo squares. Then clearly ℓ(x) ≤ 3 and we have also x ∈ D_{F} (⟨1, abc⟩) (with −a, −b and −c also −abc lies in D_{F} (⟨1, −x⟩)). It follows from (a) that ℓ(x) ≠ 2. If x is a square then ℓ(−x) = ℓ(−1) = 2^{n}. Otherwise we must have ℓ(x) = 3. Then none of a, b, c, abc can be a square. Further ℓ(−x) ≥ 2^{n} − 2 by (2.3). Thus (4.2) shows that, in a representation of −x as sum of ℓ(−x) squares over F, the partial sums of length two lie in at least 2^{n} − 4 distinct nontrivial square classes. As |D_{F} (2)/F^{×}^{2}| = 2^{n} by hypothesis, at least one of these square classes must also be represented by one of a, b, c or abc. Without loss of generality we may suppose that −x = y + at^{2} with ℓ(y) = ℓ(−x) − 2. Writing x = u^{2} + av^{2} yields 0 = x − x = y + u^{2} + a(t^{2} + v^{2}). Thus 2^{n} + 1 ≤ ℓ(y) + 3 and 2^{n} ≤ ℓ(y) + 2 = ℓ(−x). Then −x = (−1) · x ∈ D_{F} (2^{n}) implies ℓ(−x) = 2^{n}.

(c) By the above lemma there are a_{3}, …, a_{n} ∈ D_{F} (2) such that π_{D(2)} is equal to ⟨a, b, c, d⟩ ⊗ ⟨⟨a_{3}, …, a_{n}⟩⟩. Suppose now that there exists an x ∈ D_{F} (⟨a, b⟩)∩D_{F} (⟨c, d⟩). Then ⟨a, b, c, d⟩ ≅ ⟨x, abx, x, cdx⟩, which is similar to ⟨1, 1, 1, abcd⟩. Hence π_{D(2)} is similar to ⟨1, 1, 1, abcd⟩ ⊗ ⟨⟨a_{3}, …, a_{n}⟩⟩ ≅ 2^{n−1} × ⟨1⟩ ⊥ ⟨⟨abcd, a_{3}, …, a_{n}⟩⟩. It follows that the form (2^{n−1} + 1) × ⟨1⟩ is a Pfister neighbor of π_{D(2)}, hence isotropic since π_{D(2)} is hyperbolic. This is a contradiction to s(F) = 2^{n}.

(d) After multiplying by a in the statement we may suppose that a = 1. Suppose that there exists x ∈ D_{F} (⟨1, b⟩) ∩ D_{F} (⟨1, c⟩) ∩ D_{F} (⟨b, c⟩). It follows −b, −c ∈ D_{F} (⟨1, −x⟩), thus bc ∈ D_{F} (⟨1, −x⟩) ∩ D_{F} (⟨1, 1⟩) ⊂ D_{F} (⟨1, x⟩). Therefore we have ⟨1, b, c, bc⟩ ≅ ⟨1, x, bcx, bc⟩ ≅ ⟨bc, bcx, bcx, bc⟩, whence ⟨1, b, c, bc⟩ ≅ ⟨1, 1, x, x⟩. Next we choose a_{3}, …, a_{n} ∈ D_{F} (2) such that π_{D(2)} ≅ ⟨1, b, c, bc⟩ ⊗ ⟨⟨a_{3}, …, a_{n}⟩⟩ and obtain π_{D(2)} ≅ ⟨⟨1, x, a_{3}, …, a_{n}⟩⟩ ≅ 2^{n−1} × ⟨⟨x⟩⟩ ≅ 2^{n} × ⟨1⟩, since a_{3}, …, a_{n} ∈ D_{F} (2), n ≥ 3 and x ∈ D_{F} (4). This is contradictory since π_{D(2)} is hyperbolic but s(F) = 2^{n}.

(e) Let x ∈ D_{F} (⟨1, a⟩) ∩ D_{F} (⟨1, b⟩). Then, certainly, x and cx belong to D_{F} (4). If ℓ(cx) ≤ 2 then ℓ(x) ≤ 2 and (2.3) yields ℓ(−x) ≥ 2^{n} − 1. Suppose now ℓ(cx) = 3 and write cx = e + t^{2} with t ∈ F^{×} and e ∈ D_{F} (2). We have cx ∈ D_{F} (⟨c, ac⟩) ∩ D_{F} (⟨c, bc⟩) ∩ D_{F} (⟨1, e⟩). Since 1, c, ac and bc represent distinct square classes, we conclude with (c) that e and c lie in the same square class. Therefore x ∈ D_{F} (⟨1, a⟩) ∩ D_{F} (⟨1, b⟩) ∩ D_{F} (⟨1, c⟩), which by (b) implies ℓ(−x) = 2^{n}. □

4.8. Theorem. Let F be a nonreal field of level s, equal to q̄_{1}(F). Any representation (4.3) of zero as a nontrivial sum of s + 1 squares over F may be reordered in such way that the following holds: for {i, j}, {i′, j′} ∈ P_{2}^{s+1} the partial sums x_{i}^{2} + x_{j}^{2} and x_{i′}^{2} + x_{j′}^{2} lie in the same square class if and only if max{i, j, 3} = max{i′, j′, 3}.

Proof: Let G be a complete graph in s + 1 vertices v_{1}, …, v_{s+1} and with the edge-coloring given by f : P_{2}^{s+1} → D_{F} (2)/F^{×}^{2}, {i, j} ↦ (x_{i}^{2} + x_{j}^{2})F^{×}^{2} (see at the beginning of this section). We know from (4.4) that exactly s − 1 colors appear in G. Further, G does not contain any triangle with three different colors; indeed, such a triangle would correspond to a partial sum of three squares x := x_{i}^{2} + x_{j}^{2} + x_{k}^{2} with 1 ≤ i < j < k ≤ s + 1 where a := x_{i}^{2} + x_{j}^{2}, b := x_{i}^{2} + x_{k}^{2} and c := x_{j}^{2} + x_{k}^{2} lie in three distinct square classes which is impossible by part (b) of the last proposition since ℓ(−x) = s − 2. Therefore by (B.3), G is a total CC-graph. Since G has precisely (s + 1) − 2 colors we obtain from the definition of a total CC-graph in appendix B and the subsequent remarks: the vertices in G (and at the same time the x_{i}) may be renumbered in such way that for {i, j} ∈ P_{2}^{s+1} the color of the edge between v_{i} and v_{j} (i.e. the square class of x_{i}^{2}+x_{j}^{2}) depends precisely on max{i, j, 3}. □

4.9. Corollary. Let F be a nonreal field of level s = q̄_{1}(F) ≥ 8. Then q̄_{2}(F) ≥ \frac{s^{2}}{2} .

Proof: Let 0 = x_{1}^{2} + ⋯ + x_{s+1}^{2} be a representation of zero as a nontrivial sum of s + 1 squares over F. By the theorem we may, after reordering the indices, suppose that for {i, j} ∈ P_{2}^{s+1} the square class of x_{i}^{2} + x_{j}^{2} depends precisely on max{i, j, 3}.

Defining a_{i} := x_{i+1}^{2} + x_{i+2}^{2} for 1 ≤ i ≤ s−1, we get a system of representatives a_{1}, …, a_{s−1} of the s − 1 nontrivial classes of D_{F} (2)/F^{×}^{2}. Further we set c_{jk} := x_{1}^{2} + x_{j+2}^{2}+ x_{k+2}^{2} for 1 ≤ j < k ≤ s − 1. Suppose now that b c_{jk} = c_{j′k′} for b ∈ D_{F} (2) and 1 ≤ j′ < k′ ≤ s − 1. Then c_{j′k′} ∈ D_{F} (⟨1, a_{j′}⟩) ∩ D_{F} (⟨1, a_{k′}⟩) ∩ D_{F} (⟨b, b a_{j}⟩) ∩ D_{F} (⟨b, b a_{k}⟩). In view of (b), (c) and (d) of the proposition this is only possible if b ∈ F^{×}^{2}, j = j′ and k = k′. This shows that the elements c_{jk} for 1 ≤ j < k ≤ s − 1 represent distinct nontrivial classes of D_{F} (4)/D_{F} (2). Therefore q̄_{2}(F) > \binom{s−1}{2}. Since s is a power of 2, at least 8, and q̄_{2}(F) is a power of 2 or infinite we obtain q̄_{2}(F) ≥ \frac{s^{2}}{2} . □

5 Lower bounds for the square class number

We start this section with Djoković’s proof of his bound (1.6), rephrased in the terminology of appendix A.

5.1. Theorem (Djoković). If F is a nonreal field of level s ≥ 8 then

q(F) ≥ 2 · |D_{F} (s/2)/F^{×}^{2}| ≥ 2 · ∑_{i=1}^{s/2} \frac{1}{s + 2 − i} ( \begin{matrix}s + 1 \\ i \end{matrix} ) .

Proof: The first inequality is clear since |F^{×}/D_{F} (s/2)| ≥ 2. Next we consider a representation 0 = x_{1}^{2} + ⋯ + x_{s+1}^{2} of zero as a sum of s+1 nonzero squares over F. We denote by P the set of nonempty subsets of {1, …, s + 1} of cardinality not greater than s/2. We define f : P → D_{F} (s/2)/F^{×}^{2} , J ↦ (∑_{j∈J} x_{j}^{2})F^{×}^{2}. For 1 ≤ k ≤ s/2 we write f_{k} for the restriction of f to P_{k}^{s+1}. By (2.8), for k ≠ k′ the images of f_{k} and f_{k′} are disjoint. Also by (2.8), f_{k} is (k−1)-connected for any k ≤ s/2 and therefore |Im(f_{k})| ≥ \frac{1}{(s+1)−k+1} \binom{s+1}{k} by (A.4, c). All together we obtain

|D_{F} (s/2)/F^{×}^{2}| ≥ ∑_{k=1}^{s/2}|Im(f_{k})| ≥ ∑_{k=1}^{s/2} \frac{1}{s − k + 2} \binom{s + 1}{k}

which shows the second inequality. □

5.2. Remark. For an integer s ≥ 8, let ∑(s) denote the term on the right hand side in the inequality of the above theorem. Djoković showed by an elementary counting argument that ∑(s) > \frac{2^{s}}{s} [2]. As was pointed out by David B. Leep, the argument may be improved to obtain the bound ∑(s) > \frac{2^{s+1}}{s} for every even s ≥ 8. Under the hypothesis of the last theorem one has thus q(F) > \frac{2^{s+1}}{s}; further, since s = s(F) is a power of 2 and q(F) is also a power of 2 or infinite, it follows that q(F) ≥ \frac{2^{s+2}}{s}. Our calculations have shown that, at least for s a power of 2 in the range between 8 and 2^{13}, actually one has \frac{2^{s+1}}{s} < ∑(s) ≤ \frac{2^{s+2}}{s}.

However, for level 8 and 16 we get stronger bounds on q(F).

5.3. Theorem. Let F be a field. If s(F) = 8 then q(F) ≥ 512. If s(F) = 16 then q(F) ≥ 2^{15}.

Proof: Under the hypothesis s(F) = 8 we have q̄_{3}(F) ≥ 2, q̄_{2}(F) ≥ 16 (3.1) and q̄_{1}(F) ≥ 8 (1.5). Moreover, by (4.9) one of the last two inequalities must be proper. From |F^{×}/F^{×}^{2}| ≥ q̄_{1}(F)·q̄_{2}(F)·q̄_{3}(F) we get therefore q(F) ≥ 512, since F^{×}/F^{×}^{2} is an elementary abelian 2-group. For s(F) = 16 we have by the previous sections q̄_{4}(F) ≥ 2, q̄_{3}(F) ≥ 16, q̄_{2}(F) ≥ 32 and q̄_{1}(F) ≥ 16 and one of the last two inequalities must be proper. As |F^{×}/F^{×}^{2}| ≥ q̄_{1}(F) ⋯ q̄_{4}(F) this leads to q(F) ≥ 2^{15}. □

For s(F) = 2^{n} with n ≥ 5 the analogous arguments are not sufficient to improve Djoković’s result. For s(F) = 32, for example, we may get in this way q(F) ≥ 2^{25} while (5.1) yields q(F) ≥ 2^{29}.

5.4. Theorem. Let F be a field of level 2^{n} with n ≥ 3. Then |k_{n−1}F | ≥ 128. More precisely, the subgroup {−1}^{n−2}k_{1}F of k_{n−1}F is of index at least 4 and order at least 32.

Proof: Again, we use the notation ε := {−1} ∈ k_{1}F. The homomorphism F^{×} → {−1}^{n−2}k_{1}F which maps x ∈ F^{×} to the symbol ε^{n−2} · {x}, has kernel D_{F} (2^{n−2}). Since q̄_{n}(F) ≥ 2 and q̄_{n−1}(F) ≥ 16 by (3.1), we have |F^{×}/D_{F} (2^{n−2})| ≥ q̄_{n}(F) · q̄_{n−1}(F) ≥ 32. Therefore {−1}^{n−2}k_{1}F has at least 32 elements. To show that the index of this group in k_{n−1}F is at least 4 we just need to find α, β, γ ∈ k_{n−1}F \ {−1}^{n−2}k_{1}F such that α + β + γ ∈ {−1}^{n−2}k_{1}F. By the hypothesis there are a, b, c ∈ D_{F} (3 · 2^{n−3}) \ D_{F} (2^{n−2}) such that a + b + c = 0. In k_{2}F we compute {−a, −b} + {−a, −c} + {−b, −c} = {−a, bc} + {a, −bc} = {−1, abc}. Therefore we are finished if we show that none of the symbols ε^{n−3}{−a, −b}, ε^{n−3}{−a, −c} and ε^{n−3}{−b, −c} in k_{n−1}F lies actually in {−1}^{n−2}k_{1}F. If this is not true we may by case symmetry suppose that ε^{n−3}{−a, −b} = ε^{n−2}{−x} for some x ∈ F^{×}. Then the (n−1)-fold Pfister forms 2^{n−3} × ⟨⟨a, b⟩⟩ and 2^{n−2} × ⟨⟨x⟩⟩ over F are isometric, i.e. the quadratic form φ := 2^{n−3} × ⟨1, x, x, −a, −b, −ab⟩ over F is hyperbolic. It follows that any subform of φ of dimension greater than \frac{1}{2} dim(φ) = 3 · 2^{n−3} is isotropic. In particular, the form 2^{n−2} × ⟨−ax⟩ ⊥ 2^{n−3} × ⟨1⟩ ⊥ ⟨b⟩, similar to a subform of φ, must be isotropic. It follows that ax ∈ D_{F} (2^{n−2}) · D_{F} (2^{n−3} ×⟨1⟩ ⊥ ⟨b⟩) ⊂ D_{F} (2^{n−1}) whence x ∈ D_{F} (2^{n−1}). On the other hand, φ ≅ 2^{n−3} × ⟨1, x, x, c, abc, −ab⟩ shows that 2^{n−2}×⟨x⟩ ⊥ 2^{n−3}×⟨1⟩ ⊥ ⟨c⟩ is isotropic. This in turn implies that −x ∈ D_{F} (2^{n−2}) · D_{F} (2^{n−3} ×⟨1⟩ ⊥ ⟨c⟩) ⊂ D_{F} (2^{n−1}). Together this leads to −1 ∈ D_{F} (2^{n−1}) which contradicts s(F) = 2^{n}. □

5.5. Corollary. Let F be a nonreal field with s(F)≥8. Then |Br_{2}(F)| ≥ 128 and |W (F)| ≥ 2^{18}.

Proof: If s(F) = 8 then the theorem shows |k_{2}F | ≥ 128. But this is also true if s(F) = 2^{n} > 8 since then already the subgroup {−1}k_{1}F, isomorphic to F^{×}/D_{F} (2), has order at least q̄_{n}(F) · q̄_{n−1}(F) · q̄_{n−2}(F) which is sufficiently large by the results of section 3. By Merkuriev’s theorem, Br_{2}(F) is isomorphic to k_{2}F, so in particular we have |Br_{2}(F)| ≥ 128. (In fact, the arguments to estimate the size of k_{2}F work similarly for Br_{2}(F), so it is not necessary to invoke Merkuriev’s theorem here.) Let I denote the fundamental ideal of W (F) and let Ī^{i} := I^{i}/I^{i+1} for i ≥ 0. For i = 0, 1, 2 it follows from [9] that Ī^{i} ≅ k_{i}F. Thus |Ī^{0}| = 2, |Ī^{1}| = q(F) ≥ 512 and |Ī^{2}| ≥ 128. Moreover, s(F) ≥ 8 implies |Ī^{3}| ≥ 2. Therefore |W (F)| ≥ |Ī^{0}| · |Ī^{1}| · |Ī^{2}| · |Ī^{3}| ≥ 2^{18}. □

A Hypergraphs with connected colorings

In this appendix t, k and n denote nonnegative integers with t ≤ k ≤ n. We briefly say k-set for a set of cardinality k. A k-hypergraph is a system H = (V, E) where V is a set whose elements are called vertices and E a collection of distinct k-subsets of V called edges. A graph in the usual sense is then just a 2hypergraph. Let H = (V, E) be a k-hypergraph. Its number of vertices |V | is called the order of H . We say that H is complete if each k-subset of V is actually an edge, i.e. if E = {E ⊂ V | |E| = k}. By an edge-coloring of H we mean a function f : E → C. We consider the elements of C as colors and for E ∈ E we call f(E) the color of E. For t > 0 we say that the edge-coloring f is t-connected if any two edges of the same color meet in at least t vertices, i.e. if for any E, E′ ∈ E with f(E) = f(E′) we have |E ∩ E′| ≥ t.

A.1. Problem. Let t, k, n be nonnegative integers with t ≤ k ≤ n. Let H = (V, E) be a complete k-hypergraph of order n. What is the least integer m such that there exists a t-connected edge-coloring f : E → C on H with |C| = m ?

The integer m which meets the condition in the problem depends only on the values of t, k and n and will be denoted by M(t, k, n). We recall our notation P_{k}^{n} for the set of all k-subsets of {1, …, n}. A complete k-hypergraph of order n is then given by K_{k}^{n} := ({1, …, n}, P_{k}^{n}). So M(t, k, n) is just the least integer m such that there exists a function f : P_{k}^{n} → C where |C| = m and such that f(X) = f(X′) implies |X ∩ X′| ≥ t for any X, X′ ∈ P_{k}^{n}. To study M(t, k, n) as a function in t, k and n we use the theory of intersecting families in combinatorics. Let F be a family of sets. We write ⋃ F (resp. ⋂ F) for the union (resp. the intersection) of all sets belonging to F. If |U ∩ V | ≥ t holds for every U, V ∈ F then we say that the family F is t-intersecting (just intersecting for t = 1). A coloring f : E → C of a k-hypergraph H = (V, E) is thus t-connected if and only if f^{−1}({c}) is a t-intersecting family for every c ∈ C. The crucial result on intersecting families is the Erdös-Ko-Rado theorem [4] which we state in the slightly stronger version of [14]:

A.2. Theorem (Erdös-Ko-Rado). Let n ≥ (k − t + 1)(t + 1). If F is a t-intersecting family of k-subsets of an n-set then |F| ≤ \binom{n−t}{k−t}.

This theorem gives the optimal bound. Indeed, if N is an n-set and T a t-subset then F := {U ⊂ N | |U| = k, T ⊂ U} is a t-intersecting family with precisely \binom{n−t}{k−t} elements. However, under the additional condition | ⋂ F| < t, better bounds on |F| can be given. In the case t = 1 this is the following main result of [6]. (A short proof of this can be found in [5] where the case t > 1 is also treated.)

A.3. Theorem (Hilton-Milner). Let F be a family of pairwise intersecting k-subsets of an n-set such that ⋂ F = ∅. Then |F| ≤ \binom{n−1}{k−1} − \binom{n−k−1}{k−1} + 1.

Now we begin with the investigation M(t, k, n) as a function in t, k and n with 0 < t ≤ k ≤ n. We first treat the easy cases when t and k take extremal values. Part (c) is implicitly shown in [2].

A.4. Proposition. (a) M(t, k, n) = 1 is equivalent to n ≤ 2k − t.

(b) M(t, k, n) = \binom{n}{k} is equivalent to k = t.

(c) M(k−1, k, n) = M(n−k−1, n−k, n) ≥ \frac{1}{n−k+1} \binom{n}{k} for 1 ≤ k ≤ n/2.

Proof: (a) M(t, k, n) is equal to 1 if and only if P_{k}^{n} is t-intersecting; this is the case if and only if n ≤ 2k − t.

(b) Each condition holds if and only if any nonempty t-intersecting family of k-subsets of {1, …, n} consists of just one k-set.

(c) It is quite obvious that a family F ⊂ P_{k}^{n} is (k − 1)-intersecting if and only if the family of complement sets {{1, …, n} \ U | U ∈ F} is (n−k−1)- intersecting. So f : P_{k}^{n} → C is (k −1)-connected if and only if f′ : P_{n−k}^{n} → C, V ↦ f({1, …, n} \ V) is (n−k −1)-connected. This shows in particular M(k−1, k, n) = M(n−k−1, n−k, n). For a (k−1)-intersecting family F ⊂ P_{k}^{n} it is easy to check that either | ⋂ F| ≥ k−1 or | ⋃ F| ≤ k+1. In the first case we conclude |F| ≤ n − k + 1 and in the second case |F| ≤ k + 1 ≤ n − k + 1. If now f : P_{k}^{n} → C is (k−1)-connected then P_{k}^{n} is covered by the (k−1)-intersecting families f^{−1}({c}) for c ∈ C, which implies that \binom{n}{k} = |P_{k}^{n}| ≤ (n − k + 1) · |C|. □

A.5. Examples. (1) The function f : P_{k}^{n} → P_{t}^{n−k+t} which associates to X ∈ P_{k}^{n} the set of the t smallest numbers in X is a t-connected edge-coloring of K_{k}^{n}.

(2) If n≥2k−1 then a 1-connected edge-coloring of K_{k}^{n} is given by

f : P_{k}^{n} ⟶ {1, …, n−2k+2} , X ⟼ max (X ∪ {2k−1}) − 2k + 2 .

(3) Let t < k < n. If f : P_{k}^{n} → C be a t-connected edge-coloring of K_{k}^{n} and g : P_{k+1}^{n} → C′ is a (t+1)-connected edge-coloring of K_{k+1}^{n}, where C and C′ are disjoint sets, then a (t+1)-connected edge-coloring of K_{k+1}^{n+1} is defined by

h : P_{k+1}^{n+1} ⟶ C ∪ C′ , X ⟼ \{ \begin{matrix}f (X \\{n+1\}) & if n+1 ∈ X, \\ g (X) & otherwise. \end{matrix}

From these examples we conclude:

A.6. Proposition. (a) M(t, k, n) ≤ \binom{n−k+t}{t}.

(b) If n ≥ 2k − 1 then M(1, k, n) ≤ n − 2k + 2.

(c) If t<k <n then M(t+1, k+1, n+1) ≤ M(t, k, n)+M(t+1, k+1, n).

□

For lower bounds on M(t, k, n) we first consider the case t ≥ 2.

A.7. Theorem. Let 2 ≤ t < k. Then for n ≥ (k − t + 1)(t + 1) we have

M(t, k, n) ≥ ∏_{i=0}^{t−1} \frac{n − i}{k − i} > (\frac{n}{k})^{t} .

Proof: Let f : P_{k}^{n} → C be a t-connected edge-coloring of K_{k}^{n} with n ≥ (k − t + 1)(t + 1). For each c ∈ C we have then by the Erdös-Ko-Rado theorem |f^{−1}(\{c\})| ≤ \binom{n−t}{k−t}. As P_{k}^{n} = ⋃_{c∈C} f^{−1}(\{c\}) we get \binom{n}{k} ≤ |C| · \binom{n−t}{k−t}.

Therefore |C| ≥ \frac{n}{k} · \frac{n−1}{k−1} ⋯ \frac{n−t+1}{k−t+1} and an easy computation shows the second inequality. □

For the purposes of section 3 we state the following particular case:

A.8. Corollary. Let i and m be positive integers satisfying either 2≤i≤ \frac{m}{2} or 3≤i= \frac{m+1}{2} or 5≤i= \frac{m}{2} +1. Then M(2^{i−2}+1, 2^{i}, 2^{m}) > 2^{(m−i)(2^{i−2}+1)}. □

Now we come to the case t = 1.

A.9. Lemma. For k > 1 we define the polynomial

F_{k}(X) : = ∏_{i=0}^{k−1}(X −i) − k (X −2k+1) (∏_{i=1}^{k−1}(X −i) − ∏_{i=1}^{k−1}(X −k−i) + (k−1)!) .

If k ≤ n and f : P_{k}^{n} → C is such that ⋂ f^{−1}({c}) = ∅ for every c ∈ C then either |C| ≥ n − 2k + 2 or F_{k}(n) ≤ 0.

Proof: Suppose that f has the stated property. Then the Hilton-Milner theorem implies \binom{n}{k} ≤ |C| · [\binom{n−1}{k−1} − \binom{n−k−1}{k−1} + 1]. On the other hand,

(k!)^{−1} · F_{k}(n) = \binom{n}{k} − (n − 2k + 1) · [\binom{n−1}{k−1} − \binom{n−k−1}{k−1} + 1]. Thus F_{k}(n) > 0 implies |C| > (n − 2k + 1). □

A.10. Remark. The polynomial F_{k} defined in the lemma is monic of degree k. In particular, we have F_{k}(n) > 0 for all n sufficiently large. Computation for small values of k yields: F_{2}(X) = X^{2} − 7X + 18, F_{3}(X) = X^{3} − 21X^{2} + 140X − 240 and F_{4}(X) = X^{4} − 54X^{3} + 731X^{2} − 3534X + 5880. Thus we have F_{2}(n) > 0 for any n ∈ IN, F_{3}(n) > 0 for n ≥ 3 and F_{4}(n) > 0 for n ≥ 37 whereas F_{4}(36) < 0.

A.11. Theorem. For any k ≥ 1 there is a constant c_{k} ≥ 2k − 2 such that for all n ∈ IN sufficiently large we have

M(1, k, n) = n − c_{k}.

For k ≤ 3 we have, more precisely, M(1, k, n) = n − 2k + 2 for n ≥ 2k − 1.

Proof: For k = 1 there is nothing to show since M(1, 1, n) = n. For k ≥ 2 let F_{k}(X) be defined as in the lemma. By the above remark we may choose the least integer n_{k} ≥ 2k − 1 such that F_{k}(n) > 0 for all n ≥ n_{k} − 1. In particular we have n_{2} = 3 and n_{3} = 5. Let c_{k} := n_{k} − M(1, k, n_{k}). Then (A.6, b) implies c_{k} ≥ 2k − 2 and we check that equality holds for k = 2, 3. We want to prove by induction that M(1, k, n) = n − c_{k} for n ≥ n_{k}. For n = n_{k} this is trivial statement. Suppose it is true for n − 1 ≥ n_{k}. Let f : P_{k}^{n} → C be a 1-connected edge-coloring of K_{k}^{n}. If ⋂ f^{−1}({c}) = ∅ for each c ∈ C then by the lemma we have |C| ≥ n − 2k + 2 ≥ n − c_{k}. On the other hand, if there is c ∈ C such that the intersection ⋂ f^{−1}({c}) is not empty then we may suppose that it contains the element n. Then the restriction f′ : P_{k}^{n−1} → C \ {c} of f to P_{k}^{n−1} is a 1-connected edge-coloring of K_{k}^{n−1}. By the induction hypothesis we have |C \ {c}| ≥ M(1, k, n − 1) = (n − 1) − c_{k} and thus |C| ≥ n − c_{k}. This implies M(1, k, n) ≥ n − c_{k}. But (A.6, c) shows M(1, k, n) ≤ M(1, k, n−1)+M(0, k−1, n−1) = n−c_{k} since M(0, k−1, n−1) = 1. Hence M(1, k, n) ≥ n − c_{k} which finishes the induction step. □

A.12. Question. Does M(1, k, n) = n − 2k + 2 hold for all n ≥ 2k − 1, even if k > 3 ?

B CC-Graphs

In this appendix we study connected edge-colorings for usual complete graphs. Here we are not only interested in the minimal number of colors but also in the distribution of the colors in the graph.

Let G denote a complete graph of order n with vertices v_{1}, …, v_{n} and colored edges. The distribution of colors in G can be equivalently represented by an edge-coloring of K_{2}^{n} (see appendix A), i.e. by a function f : P_{2}^{n} → C, where C stands for the set of colors in G and f associates to {i, j} ∈ P_{2}^{n} the color of the edge between the vertices v_{i} and v_{j}. A set of all the edges of a certain color shall be called a color-component. If such a color-component consists of r ≥ 3 edges all together having a vertex x in common we call it an r-star and x its center. By a triangle in G we mean a complete subgraph of order 3 of G. A triangle is said to be monochrome (resp. three-colored) if the three edges are of the same color (resp. of three different colors). A second complete colored graph G′ of order n is said to be equivalent to G if there is a bijection between the sets of vertices of G and G′ such that the induced bijection on the sets of edges preserves the color-components (in both directions). We call G color-connected or a CC-graph if in G any two edges of the same color are adjacent. This is equivalent to the edge-coloring f being 1-connected. The only possible color-components in G are then single edges, pairs of edges with a vertex in common, stars and monochrome triangles. Theorem (A.11) says that M(1, 2, n) = n − 2 for n ≥ 3. This corresponds to a result of [13]. We rephrase it as follows and give a direct proof.

B.1. Proposition (Tort). A CC-graph of order n ≥ 3 has at least n − 2 colors.

Proof: For n = 3 the statement is trivial. If n > 3 and G has less than n colors then one of its color-components must be a star. Deleting the center of this star yields a CC-graph G′ of order n − 1 with less colors. By induction hypothesis G′ has at least n − 3 and therefore G at least n − 2 colors. □

For any n ≥ 3 the complete graph K_{2}^{n}, whose vertices are the integers 1, …, n, together with the 1-connected coloring f_{n} : P_{2}^{n} → {1, …, n − 2}, {i, j} ↦ max{i, j, 3}−2 defines a particular CC-graph G_{n} of order n with n−2 colors (compare with example (A.5, 2)). The color-components of G_{n} are one monochrome triangle and one i-star for each 3 ≤ i ≤ n−1. For 3 ≤ n ≤ 5, every CC-graph with n−2 colors is equivalent to G_{n}. This is not true for n = 6, since there is a CC-graph of order 6 with color-components a triangle and three 4-stars.

B.2. Proposition. Let G be a CC-graph with n ≥ 3 vertices and n − 2 colors. Then G has as color-components one monochrome triangle and n − 3 stars. Moreover, each vertex of G lies either on the monochrome triangle or is the center of exactly one star.

Proof: Let G′ be the complete subgraph spanned by all vertices of G which are not the center of a star in G. We want to show that G′ is a monochrome triangle. Then the vertices of G outside of G′ will be the centers of n − 3 stars and as G has just n − 2 colors the entire statement follows.

Let n′ be the order of G′. The n − n′ vertices of G outside of G′ are all centers of stars whose colors do not appear in G′. As a consequence, G′ has at least n − n′ colors less than G. Then by (B.1), G′ has exactly n′ − 2 colors. Since G′ is a graph without stars each color appears at most three times, counting the edges yields 3(n′ − 2) ≥ \frac{n′(n′−1)}{2} whence n′ ≤ 5. As G′ has n′ − 2 colors and contains no star, we have n′ = 3 and G′ is a monochrome triangle. □

A CC-graph G will be called total if there is a permutation σ ∈ S_{n} such that for any {i, j} ∈ P_{2}^{n} the color of the edge between v_{i} and v_{j} depends only on max{σ(i), σ(j)}. After renumbering the vertices G we may then suppose that the permutation σ is the identity on {1, …, n}. Let G be a total CC-graph of order n with vertices v_{1}, …, v_{n} enumerated in such a way that the color of any edge linking v_{i} and v_{j} depends only on max{i, j}. Then G has at most n−1 different colors. From (B.1) it follows that the number of colors in G is either n − 2 or n − 1. Further, by (B.2) the number of colors is n − 2 if and only if v_{1}, v_{2} and v_{3} form a monochrome triangle and then the color of the edge between v_{i} and v_{j} depends precisely on max{i, j, 3}. In both cases the enumeration of the vertices is unique up to changing the first three respectively the first two indices. Moreover, G contains exactly n − 3 stars. More precisely, for each 4 ≤ i ≤ n there is exactly one (i−1)-star in G whose center is v_{i}. It is clear from the definition that a complete subgraph of a total CC-graph is also a total CC-graph.

B.3. Proposition. A CC-graph G is total if and only if it contains no threecolored triangle.

Proof: The necessity of the condition follows from the definition of a total CC-graph. Suppose now that G is a CC-graph with n vertices with no threecolored triangle. We show by induction on n that G is total. For n ≤ 3 this is evident. If n ≥ 4 then any complete subgraph with 4 vertices contains a star since otherwise it would contain a three-colored triangle. So we can choose an r-star in G where r is as large as possible. For the ease of imagination say, it is of red color. We may suppose that v_{n} is the center of this star. Let G′ be the complete subgraph of G with all the vertices of G except v_{n}. Then G′ is also a CC-graph with n − 1 vertices and contains no three-colored triangle. So, by the induction hypothesis, G′ is total, i.e. its vertices can be enumerated as v_{1}, …, v_{n−1} in such a way that the color of an edge connecting vertices v_{i} and v_{j} depends just on max{i, j}. This would still be true for the enumeration of the vertices v_{1}, …, v_{n} of G, if v_{n} is connected with each of the v_{1}, …, v_{n−1} by an edge of red color. So we just have to show that r = n − 1. Suppose that r < n − 1. Then certainly n > 4 since r ≥ 3 by the definition of an r-star. But v_{n−1} is the center of an n−2-star in G′, say of blue color. By the maximality of r we see that the edge between v_{n−1} and v_{n} cannot be blue and that r = n − 2. So there must be exactly one vertex v_{k} with 1 ≤ k ≤ n − 1 which is connected with v_{n} with an edge of color different from red. It cannot be of blue color either so say that its color is green. Now we see that there is a triangle of colors red, blue and green contained in G, formed by v_{k}, v_{n−1}, v_{n} if k < n − 1 and by v_{1}, v_{n−1}, v_{n} if k = n − 1, which gives the desired contradiction. □

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Karim Johannes Becher

Laboratoire de Mathématiques U.F.R. Sciences et Techniques Université de Franche-Comté 16, Route de Gray 25030 Besançon Cedex France becher@math.univ-fcomte.fr