Homology, Homotopy and Applications, vol.6(1), 2004, pp.17–31

SYMBOL LENGTHS IN MILNOR K-THEORY

KARIM JOHANNES BECHER and DETLEV W. HOFFMANN

(communicated by Ulf Rehmann)

Abstract Let F be a field and p a prime number. The p-symbol length of F, denoted by λ_{p}(F), is the least integer l such that every element of the group K_{2}F/pK_{2}F can be written as a sum of ⩽ l symbols (with the convention that λ_{p}(F) = ∞ if no such integer exists). In this article, we obtain an upper bound for λ_{p}(F) in the case where the group F^{×}/F^{×}^{p} is finite of order p^{m}. This bound is λ_{p}(F) ⩽ \frac{m}{2}, except for the case where p = 2 and F is real, when the bound is λ_{2}(F) ⩽ \frac{m+1}{2}. We further give examples showing that these bounds are sharp.

1. Introduction

Let F be a commutative field. Let F^{×} denote the multiplicative group of F. We recall the definition of the groups K_{n}F (n ⩾ 1) of Milnor K-theory, as defined in [15]. K_{1}F is nothing but the multiplicative group F^{×} in additive notation. In order to distinguish between addition in K_{1}F and in F, we write {x} ∈ K_{1}F for x ∈ F^{×}, so that we have {x} + {y} = {xy} in K_{1}F for x, y ∈ F^{×}. For n ⩾ 2, the group K_{n}F is defined as the quotient of the group (K_{1}F)^{⊗n} modulo the subgroup generated by the elementary tensors {x_{1}} ⊗ ⋯ ⊗ {x_{n}} ∈ (K_{1}F)^{⊗n} where x_{1}, …, x_{n} ∈ F^{×} and x_{i} + x_{i+1} = 1 for some i < n.

Let now be n ⩾ 1 and let H denote the group K_{n}F or some quotient of it. For x_{1}, …, x_{n} ∈ F^{×}, we denote the canonical image of {x_{1}} ⊗ ⋯ ⊗ {x_{n}} ∈ (K_{1}F)^{⊗n} in H by {x_{1}, …, x_{n}} and we call such an element a symbol in H. This notation differs from Milnor’s original one where the same symbol is denoted by l(x_{1}) ⋯ l(x_{n}). Obviously, H is generated by its symbols. Note further that the zero element of H is a symbol. We may then ask whether there exists an integer l ⩾ 0 such that every element of H can be written as a sum of l symbols. If this is the case then we denote by λ(H) the least such integer l; otherwise we set λ(H) = ∞. We call the value λ(H) ∈ ℕ ∪ {∞} the symbol length of H.

Quotients of K_{n}F of particular interest are K_{n}F/l K_{n}F where l is a positive integer; we shall abbreviate this quotient by K_{n}^{(l)}F. In the case where l = 2, we may also use Milnor’s notation k_{n}F for K_{n}F/2 K_{n}F.

Received September 8, 2003, revised January 13, 2004; published on February 6, 2004. 2000 Mathematics Subject Classification: Primary: 19D45, Secondary: 11E04, 11E81, 16K50. Key words and phrases: Milnor K-groups, symbols, symbol length, power norm residue algebra, symbol algebra, quadratic forms, level, Brauer group, Merkurjev-Suslin theorem. \begin{matrix}c \\ ◯ \end{matrix} 2004, Karim Johannes Becher and Detlev W. Hoffmann. Permission to copy for private use granted.

Let now p be a prime number. We define λ_{p}(F) = λ(K_{2}^{(p)}F) and call this the p-symbol length of the field F.

Lenstra showed that if F is a global field, then K_{2}F consists of symbols (cf. [10]); it follows then that λ_{p}(F) = 1 for all primes p. The same is true if F is a local field. Whether the rational function field in two variables over the complex numbers F = ℂ(X, Y) also satisfies λ_{p}(F) = 1 for all primes p is a striking open question; at least for p ⩽ 3 the answer is positive, by a theorem of Artin (cf. [1]). A recent result of Saltman implies that λ_{2}(F) = 2 if F is the function field of a curve over a q-adic local field where q ≠ 2 (cf. [19]).

In this article, we establish an upper bound for λ_{p}(F) under the assumption that F^{×}/F^{×p} is finite and we show further that it is generally the best possible.

Theorem 1.1. Suppose that |F^{×}/F^{×}^{p}| = p^{m} where m ∈ ℕ. Then

λ_{p}(F) ⩽ \{ \begin{matrix}[\frac{m}{2}] & if p ≠ 2 or if F is nonreal, \\ [\frac{m+1}{2}] & if F is real and p = 2. \end{matrix}

Here we use the notation [x] for the integral part of x ∈ ℝ. Recall that the field F is nonreal if and only if −1 is a sum of squares in F.

The proof of Theorem 1.1 is divided into several parts, which will occupy the next two sections. In Section 2, we relate the K-groups modulo p to certain exterior power spaces and, under the condition that p ≠ 2 or that −1 is a square in F, we deduce the estimate λ_{p}(F) ⩽ [\frac{m}{2}] from a known fact on alternating spaces (2.3). In a similar way we get the weaker bound λ_{p}(F) ⩽ [\frac{m+1}{2}] for p = 2 (2.8). We will therefore be left with the task to exclude that λ_{2}(F) be equal to \frac{m+1}{2} unless F is a real field. This will be done in the third section by an argument involving quadratic form theory (3.5).

Finally we show in the fourth section that in all cases, according to whether F is real or not, and whether p equals 2 or not, the estimate stated above is best possible (4.2). To do so, we give examples where F is a field containing a primitive p-th root of unity, such that |F^{×}/F^{×}^{p}| = p^{m} and such that there is a simple F -division algebra which is a product of l symbol algebras of degree p, where l = [\frac{m+1}{2}] if p = 2 and if F is real, and where l = [\frac{m}{2}] otherwise; the existence of such a division algebra implies indeed that λ_{p}(F) ⩾ l. We will further apply a similar argument to show that for the rational function field in m + 1 variables over a field F containing a primitive p-th root of unity, we always have λ_{p}(F (X_{0}, …, X_{m}) ) ⩾ m (4.5). This improves the bound λ_{p}(F (X_{0}, …, X_{m}) ) ⩾ [\frac{m+1}{2}], shown in [7, Proposition 3] (under a stronger hypothesis).

2. K-groups modulo p and exterior powers

Let k denote a commutative field, V a vector space over k and n a positive integer. Let Λ^{n}V denote the exterior power of degree n over V. This is a vector space over k generated by elements v_{1} ∧⋯∧v_{n}, where v_{1}, …, v_{n} ∈ V, subject to the relations of k-multilinearity as well as to the relation that v_{1} ∧ ⋯ ∧ v_{n} = 0 whenever v_{i} = v_{i+1} for some i < n. An element of Λ^{n}V which is of the shape v_{1} ∧ ⋯ ∧ v_{n} is called a pure n-vector.

Suppose now that V has finite dimension m. There exists a least integer N such that every element of Λ^{n}V is a sum of N pure n-vectors. If the dimension of V is m then we denote this integer by N(k; m, n) (since it depends only on k, n and m). For n = 2 a classical result tells us that N(k; m, 2) = [\frac{m}{2}], independently of the field k (cf. [3, §5, Corollaire 2]).

Let p be a prime number and n a positive integer. Let 𝔽_{p} denote the finite field with p elements. The group K_{n}^{(p)}F, associated to the field F, is endowed with a natural structure as a vector space over 𝔽_{p}. Note that for m ∈ ℕ, equality |F^{×}/F^{×}^{p}| = p^{m} means that K_{1}^{(p)}F as a vector space over 𝔽_{p} has dimension m.

Let x_{1}, …, x_{n} be elements of F^{×}. If x_{i} = x_{i+1} for some i < n then one has

{x_{1}, …, x_{n}} = {x_{1}, …, x_{i}, −1, x_{i+2}, …, x_{n}}

in K_{2}^{(p)}F by [15, Lemma 1.2]; in particular, this symbol is of order at most 2. Suppose now that −1 ∈ F^{×}^{p}, that is either p ≠ 2 or p = 2 and −1 ∈ F^{×}^{2}. This implies that any symbol in K_{n}^{(p)}F with a repeated coefficient is zero. Therefore, if −1 ∈ F^{×p} then the natural 𝔽_{p}-homomorphism (K_{1}^{(p)}F)^{n} ⟶ K_{n}^{(p)}F induces a natural 𝔽_{p}-homomorphism

Λ^{n}(K_{1}^{(p)}F) ⟶ K_{n}^{(p)}F

which maps a pure n-vector {x_{1}} ∧ ⋯ ∧ {x_{n}} to the symbol {x_{1}, …, x_{n}} and which is therefore surjective. From this we conclude:

Lemma 2.1. Suppose that p ≠ 2 or that −1 ∈ F^{×}^{2}. If |F^{×}/F^{×}^{p}| = p^{m} then λ(K_{n}^{(p)}F) ⩽ N(𝔽_{p}; m, n). □

Unfortunately, the exact value of N(𝔽_{p}; m, n) is not known in general when n ⩾ 3 (cf. [6] on this problem and approximative results).

We may further observe that the bound in (2.1) is best possible for all m, n and p. Indeed, one will have λ(K_{n}^{(p)}F) = N(𝔽_{p}; m, n) if −1 ∈ F^{×}^{p}, |F^{×}/F^{×}^{p}| = p^{m} and if the natural 𝔽_{p}-homomorphism Λ^{n}(K_{1}^{(p)}F) ⟶ K_{n}^{(p)}F is bijective, and examples where these conditions hold will be given in (2.5).

Proposition 2.2. Suppose that p ≠ 2 or that −1 ∈ F^{×}^{2}. The following are equivalent:

(i) for any n ⩾ 2, the natural 𝔽_{p}-homomorphism Λ^{n}(K_{1}^{(p)}F) ⟶ K_{n}^{(p)}F is an isomorphism;

(ii) the natural 𝔽_{p}-homomorphism Λ^{2}(K_{1}^{(p)}F) ⟶ K_{2}^{(p)}F is an isomorphism;

(iii) whenever a, b ∈ F^{×} are such that the symbol {a, b} ∈ K_{2}^{(p)}F is zero then the

elements {a} and {b} in K_{1}^{(p)}F are 𝔽_{p}-linearily dependent; (iv) for any a ∈ F^{×} \ F^{×}^{p} there exists r ⩾ 0 such that a^{r}(1 − a) ∈ F^{×}^{p}.

Proof. The implications (i ⇒ ii ⇒ iii ⇒ iv) are obvious.

Assume now that (iv) holds. Then it follows for n ⩾ 2 that a pure n-vector {x_{1}}∧⋯∧{x_{n}} ∈ Λ^{n}(K_{1}^{(p)}F) is zero as soon as x_{i} +x_{i+1} = 1 for some i < n. Since this corresponds to the defining relation for K_{n}^{(p)}F as a quotient of (K_{1}^{(p)}F)^{⊗n}, we conclude that the natural surjective 𝔽_{p}-homomorphism Λ^{n}(K_{1}^{(p)}F) ⟶ K_{n}^{(p)}F is in fact bijective. This shows that (iv) implies (i). □

In the case where p ≠ 2 and F contains a primitive p-th root of unity, the conditions (i)–(iv) are satisfied if and only if F is a p-rigid field, as defined in [23, p. 772].

Proposition 2.3. Suppose that |F^{×}/F^{×}^{p}| = p^{m}, and further that p ≠ 2 or that −1 ∈ F^{×}^{2}. Then

(a) every element of K_{2}^{(p)}F is a sum of [\frac{m}{2}] symbols,

(b) every element of K_{m−1}^{(p)}F is a symbol,

(c) K_{n}^{(p)}F = 0 for any n > m,

(d) the equivalent conditions in (2.2) hold if and only if K_{m}^{(p)}F ≠ 0, and in this case one has K_{m}^{(p)}F ≅ ℤ/p ℤ.

Proof. (a) By (2.1) every element of K_{2}^{(p)}F can be written as a sum of N(𝔽_{p}; m, 2) symbols, and this number is known to be equal to [\frac{m}{2}].

(b) It is easy to see that N(k; m, m−1) = 1 for any field k. Therefore we have λ(K_{m−1}^{(p)}F) ⩽ 1 by (2.1).

(c) K_{n}^{(p)}F can be considered as a quotient of the space Λ^{n}(K_{1}^{(p)}F), which vanishes as soon as n exceeds m, the dimension of K_{1}^{(p)}F.

(d) Since K_{1}^{(p)}F is of dimension m, one has Λ^{m}(K_{1}^{(p)}F) ≅ ℤ/p ℤ. Therefore the first condition in Proposition 2.2 implies that K_{m}^{(p)}F ≅ ℤ/p ℤ.

Assume now that the equivalent conditions in Proposition (2.2) do not hold for F. In particular, since (iii) does not hold, there are a_{1}, a_{2} ∈ F^{×}, 𝔽_{p}-linearly independent modulo p-th powers, such that {a_{1}, a_{2}} = 0 in K_{2}^{(p)}F. We may choose a_{3}, …, a_{m} ∈ F^{×} such that {a_{1}}, …, {a_{m}} form an 𝔽_{p}-basis of K_{1}^{(p)}F. Then the pure n-vector {a_{1}}∧⋯∧{a_{m}} in Λ^{m}(K_{1}^{(p)}F) is nonzero, while the symbol {a_{1}, …, a_{m}} in K_{m}^{(p)}F is zero. Hence the natural surjection Λ^{m}(K_{1}^{(p)}F) ⟶ K_{m}^{(p)}F is not injective. As Λ^{m}(K_{1}^{(p)}F) ≅ ℤ/p ℤ, it follows that K_{m}^{(p)}F = 0. □

Lemma 2.4. Suppose p ≠ 2 or that −1 ∈ F^{×}^{2}, and that F is p-henselian with respect to a discrete valuation v with residue field F̄ with char(F̄) ≠ p. If the equivalent conditions in (2.2) hold for F̄ then they hold for F as well.

Proof. Since F is p-henselian and p ≠ char(F̄), we know that for any u ∈ F with v(u) = 0 we have u ∈ F^{×}^{p} if and only if \overline{u} ∈ F̄^{×p}. We show that Condition (iv) in (2.2) holds for the field F if it holds for F̄.

Let a be an element of F which is not a p-th power in F. We are looking for an integer r such that a^{r} −a^{r+1} is a p-th power in F. Using that −1 ∈ F^{×}^{p} and possibly replacing a by −a^{−1}, we may assume that v(a) ⩾ 0. If v(a) > 0 then a^{r} − a^{r+1} is a p-th power if and only of a^{r} is a p-th power, hence we are done with r = p. On the other hand, if v(a) = 0 then we saw that \overline{a} cannot be a p-th power in F̄; hence, by hypothesis, there exists r ∈ ℤ such that \overline{a^{r}(1 − a)} is a p-th power in F̄ and it follows that a^{r}(1 − a) is a p-th power in F. □

Examples 2.5. In each of the following cases, the field F satisfies the equivalent conditions of (2.2):

• F = C((X_{1})) … ((X_{m})) where C is an algebraically closed field,

• F = R((X_{1})) … ((X_{m})) where p ≠ 2 and R is a real closed field,

• F is a non-p-adic m-dimensional local field, i.e., there exists a sequence of fields F_{0}, …, F_{m} = F where F_{0} is finite with char(F_{0}) ≠ p and, for 1 < i ⩽ m, the field F_{i} is complete with respect to a discrete valuation with residue field F_{i−1}.

Indeed, this follows from the basic case m = 0, using the above lemma. Note further that |F^{×}/F^{×}^{p}| = p^{m} in the first two cases. In the third case one has |F^{×}/F^{×}^{p}| = p^{m+1} if p divides (char(F_{0}) − 1) = |F_{0}^{×}| and |F^{×}/F^{×}^{p}| = p^{m} otherwise.

The remainder of this section together with the following section is devoted to the study of the case where p = 2. Then, without the hypothesis that −1 ∈ F^{×}^{2}, we cannot relate K_{n}^{(p)}F to the exterior power space Λ^{n}(K_{1}^{(p)}F) over 𝔽_{p} as before. We therefore are going to define a kind of ”twisted exterior power space” over the field with two elements.

In the sequel, the letter k will only be used to denote the Milnor K-groups modulo 2 of a field, that is k_{n}F = K_{n}^{(2)}F (n ⩾ 0).

Let V be a vector space over the field 𝔽_{2}, ε a fixed element of V and n a positive integer. We shall write Λ_{ε}^{n}V for the vector space over 𝔽_{2} generated by elements v_{1} ∧ ⋯ ∧ v_{n}, with v_{1}, …, v_{n} ∈ V, subject to the relations of multilinearity and symmetry (i.e. v_{1} ∧ ⋯ ∧ v_{n} does not depend on the order of the coefficients v_{i}) and further to the relation that v_{1} ∧ ⋯ ∧ v_{n} = 0 whenever v_{i+1} = v_{i} + ε for some i < n. Note that if ε = 0, then Λ_{ε}^{n}V is just the exterior power space Λ^{n}V.

We call pure n-vectors in Λ_{ε}^{n}V the elements of the form v_{1} ∧ ⋯ ∧ v_{n}. From now on we restrict to the case where n = 2. For given x ∈ V, we shall say that a pure 2-vector is of the shape x ∧ ∗ if it can be written as x ∧ y for some y ∈ V.

Lemma 2.6. Let V be a vector space over 𝔽_{2} with a special element ε ∈ V. Let ξ be an element of Λ_{ε}^{2}V and let v_{1}, …, v_{l}, w_{1}, …, w_{l} ∈ V be such that

ξ = v_{1} ∧ w_{1} + ⋯ + v_{l} ∧ w_{l} .

(a) If v is a nontrivial sum of some of the elements v_{1}, …, v_{l} then ξ can be written as a sum of l pure 2-vectors where the first is of the shape v ∧ ∗.

(b) If v is a nontrivial sum of some of the elements v_{1}, …, v_{l}, w_{1}, …, w_{l} then ξ can be written as a sum of l pure 2-vectors where the first is of the shape v ∧ ∗ or (v + ε) ∧ ∗.

(c) If v_{1}, …, v_{l}, w_{1}, …, w_{l}, ε are linearly dependent then ξ can be written as a sum of l pure 2-vectors where the first is of the shape ε ∧ ∗.

Proof. (a) If l = 1 this is trivial. In the case l = 2 one uses the equality

v_{1} ∧ w_{1} + v_{2} ∧ w_{2} = (v_{1} + v_{2}) ∧ w_{1} + v_{2} ∧ (w_{1} + w_{2}) .

The general case follows from this by induction on l.

(b) Using the relations

v_{i} ∧ w_{i} = w_{i} ∧ v_{i} = (v_{i} + w_{i} + ε) ∧ w_{i}

one readily sees that the statement follows from (a).

(c) Suppose that v_{1}, …, v_{l}, w_{1}, …, w_{l}, ε are linearly dependent. Then one of the elements 0 and ε can be written as a nontrivial sum of some of the elements v_{1}, …, v_{l}, w_{1}, …, w_{l}. Observing that a pure 2-vector of the shape 0∧∗ is zero, hence equal to ε ∧ 0, we conclude by (b) that ξ can be written as a sum of l pure 2-vectors where the first is of the shape ε ∧ ∗. □

Proposition 2.7. Let V be a vector space over 𝔽_{2} of dimension m and let ε ∈ V. Then any element of Λ_{ε}^{2}V is a sum of [\frac{m+1}{2}] pure 2-vectors. If ξ ∈ Λ_{ε}^{2}V is not a sum of [\frac{m}{2}] pure 2-vectors, then

ξ = ε ∧ ε + v_{1} ∧ w_{1} + ⋯ + v_{n} ∧ w_{n} ,

where v_{1}, w_{1}, …, v_{n}, w_{n} ∈ V and 2n+1 = m; moreover, under these circumstances, (v_{1}, w_{1}, …, v_{n}, w_{n}, ε) is a basis of V.

Proof. Suppose that ξ ∈ Λ_{ε}^{2}V is a sum of l but not of l − 1 pure 2-vectors where l > \frac{m}{2}. In a representation of ξ as a sum of l pure 2-vectors, the 2l coefficients are necessarily linearly dependent. By (2.6(c)), we may write ξ = ε ∧ w + ξ′ with w ∈ V and with ξ′ ∈ Λ_{ε}^{2}V equal to a sum of l −1 pure 2-vectors. Assume that w is different from ε. Then, for dimension reasons, at least one of the elements 0, ε, w and w + ε can be written as a nontrivial sum of the coefficients of any representation of ξ′ as a sum of l − 1 pure 2-vectors. Using the fact that 0 ∧ ∗ = ε ∧ 0, it follows from (2.6(b)) that ξ′ can be written as a sum of l − 1 pure 2-vectors where the first one is of the shape x ∧ y with y ∈ V and x equal to one of the elements w, ε and w + ε. Now ε ∧ w + x ∧ y cannot be equal to a pure 2-vector, since otherwise ξ would be equal to a sum of l −1 pure 2-vectors. Hence, by the relations in Λ_{ε}^{2}V, we must have x = w + ε and therefore

ε ∧ w + x ∧ y = ε ∧ (x + ε) + y ∧ x = ε ∧ ε + (y + ε) ∧ x .

From this we conclude that we can write

ξ = ε ∧ ε + v_{1} ∧ w_{1} + ⋯ + v_{l−1} ∧ w_{l−1}

with v_{1}, w_{1}, …, v_{l−1}, w_{l−1} ∈ V. Since ξ is not a sum of less than l pure 2-vectors, we conclude from (2.6(c)) that v_{1}, w_{1}, …, v_{l−1}, w_{l−1}, ε are linearly independent. Then, since 2l −1 ⩾ m, these elements form a basis of V, in particular, if we put n = l −1, then 2n + 1 = m. □

Since k_{1}F has an obvious structure of a vector space over 𝔽_{2}, we may apply the above results to k_{1}F together with the special element ε = {−1}. By the relations defining k_{n}F, there exists a surjective 𝔽_{2}-homomorphism

Λn_{{−1}}(k_{1}F) ⟶ k_{n}F

which maps a pure n-vector {x_{1}} ∧ ⋯ ∧ {x_{n}} to a symbol {x_{1}, …, x_{n}}. From this together with the last proposition we obtain immediately:

Corollary 2.8. Suppose that |F^{×}/F^{×}^{2}| = 2^{m}. Then every element of k_{2}F can be written as a sum of [\frac{m+1}{2}] symbols. □

Remark 2.9. Note that (2.1) and (2.8) immediately imply following weaker version of Theorem 1.1: if |F^{×}/F^{×}^{p}| = p^{m} then

λ_{p}(F) ⩽ \{ \begin{matrix}[\frac{m}{2}] & if p ≠ 2 or −1 ∈ F^{×}^{2}, \\ [\frac{m+1}{2}] & if p = 2 and −1 ∉ F^{×}^{2}. \end{matrix}

In order to establish Theorem 1.1 entirely, it remains to show that λ_{2}(F) ⩽ [\frac{m}{2}] holds for any nonreal field F with |F^{×}/F^{×}^{2}| = 2^{m}. This will be accomplished with (3.5) in the following section.

3. K-groups modulo 2 and quadratic forms

For the following definitions and facts, we suppose that the characteristic of F is different from 2. We shall use the standard notations in quadratic form theory as established in [8] and [20]. However, we use a different convention for Pfister forms: if a_{1}, …, a_{n} ∈ F^{×} then ⟨⟨a_{1}, …, a_{n}⟩⟩ will denote the n-fold Pfister form ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩. Let W (F) denote the Witt ring of F and IF the fundamental ideal consisting of the classes of quadratic forms of even dimension. Let further I^{n}F = (IF)^{n} and Ī^{n}F = I^{n}F/I^{n+1}F for any n ⩾ 0. The ideal I^{n}F is additively generated by the n-fold Pfister forms over F.

Milnor has defined for any n ⩾ 1 a homomorphism s_{n} : k_{n}F ⟶ Ī^{n}F, mapping a symbol {x_{1}, …, x_{n}} to the class of the Pfister form ⟨⟨x_{1}, …, x_{n}⟩⟩ and determined by this property. Milnor showed that s_{2} : k_{2}F → Ī^{2}F is an isomorphism [15, Theorem 4.1.]. It has been proven recently that s_{n} : k_{n}F ⟶ Ī^{n}F is an isomorphism for any n ⩾ 1 [18, Theorem 4.1].

We write ±F^{×}^{2} for the union F^{×}^{2} ∪ −F^{×}^{2}. Recall that F is a real euclidean field, if F^{×}^{2} is an ordering of F, i.e. if F is real and F^{×} = ±F^{×}^{2}.

Proposition 3.1. Suppose that |F^{×}/F^{×}^{2}| = 2 and n ⩾ 2. If F is nonreal then k_{n}F = 0, otherwise F is real euclidean and the unique nonzero element in k_{n}F is {−1, …, −1}.

Proof. By the hypothesis we have F^{×} = F^{×}^{2} ∪ aF^{×}^{2} for some a ∈ F^{×}. Suppose that k_{n}F is nontrivial. It is obvious that then the unique nonzero element of k_{n}F is the symbol {a, …, a}, which can also be written as {−1, …, −1, a}. As this symbol is nontrivial, −1 cannot be a square and a cannot be a sum of two squares in F.

Hence aF^{×}^{2} = −F^{×}^{2} and −1 is not a sum of two squares. Therefore, F^{×} = ±F^{×}^{2} and every sum of two squares is again a square in F. We conclude that F is a real euclidean field. To complete the proof we remind that, if F is any real field, then the symbol {−1, …, −1} ∈ k_{n}F is nontrivial. □

In the sequel we will focus on the study of k_{2}F. If φ is a quadratic form in I^{2}F then we write \overline{φ} for its class in Ī^{2}F. The following statement is folklore, at least in the context where k_{2}F is replaced by Br_{2}(F), the 2-torsion of the Brauer group of F, and s_{2} : k_{2}F → Ī^{2}F by the inverse of the homomorphism c : Ī^{2}F → Br_{2}(F), induced by the Clifford invariant (cf. [5, Lemma 2.2.] and the references given there). We wish to give a self-contained proof here in our setting, where the statement does not depend on Merkurjev’s result [11] that c : Ī^{2}F → Br_{2}(F) is an isomorphism.

Lemma 3.2. Suppose that char(F) ≠ 2. Let m ⩾ 1 and ξ ∈ k_{2}F. A necessary and sufficient condition that ξ be a sum of m symbols is that the class s_{2}(ξ) ∈ Ī^{2}F contain a quadratic form of dimension 2m + 2.

Proof. We first prove by induction on m that the condition is necessary. We write ξ = ξ′ + {a, b} where ξ′ ∈ k_{2}F is a sum of m − 1 symbols and a, b ∈ F^{×}. Now, if m = 1 then we have ξ′ = 0 and ξ = \{a, b\}, hence s_{2}(ξ) = \overline{⟨⟨a, b⟩⟩}, and the dimension of ⟨⟨a, b⟩⟩ is 4 = 2m + 2. Suppose now that m > 1. By induction hypothesis we have s_{2}(ξ′) = \overline{φ′} for a quadratic form φ′ of dimension 2(m−1) + 2 = 2m. We decompose φ′ into φ′′ ⊥ ⟨r⟩, where φ′′ is a quadratic form of dimension 2m − 1 and r ∈ F^{×}. As s_{2}(\{a, b\}) = \overline{−r⟨⟨a, b⟩⟩} we obtain

s_{2}(ξ) = s_{2}(ξ′) + s_{2}(\{a, b\}) = \overline{φ′ ⊥ −r⟨⟨a, b⟩⟩} .

But the form φ′ ⊥ −r⟨⟨a, b⟩⟩ is Witt equivalent to φ′′ ⊥ ⟨ra, rb, −rab⟩. Hence, if we put φ = φ′′ ⊥ ⟨ra, rb, −rab⟩ then we have s_{2}(ξ) = \overline{φ} and dim φ = 2m + 2.

To show that the condition is sufficient, we use again induction on m. Let ψ be a quadratic form in I^{2}F of dimension 2m + 2 such that s_{2}(ξ) = \overline{ψ}. We decompose ψ into ψ′ ⊥ ⟨c, d, e⟩ with ψ′ a quadratic form of dimension 2m − 1 and c, d, e ∈ F^{×}. Since ψ is Witt equivalent to ψ′ ⊥ ⟨−cde⟩ ⊥ e⟨⟨−ce, −de⟩⟩, we have

s_{2}(ξ) = \overline{ψ′ ⊥ ⟨−cde⟩} + \overline{e⟨⟨−ce, −de⟩⟩} = \overline{ψ′ ⊥ ⟨−cde⟩} + s_{2}(\{−ce, −de\}) .

We write ξ = ξ′ + {−ce, −de}, with ξ′ ∈ k_{2}F. Since s_{2} is an isomorphism, it follows that s_{2}(ξ′) = \overline{ψ′ ⊥ ⟨−cde⟩}, where the quadratic form ψ′ ⊥ ⟨−cde⟩ has dimension 2m = 2(m − 1) + 2 and trivial discriminant. Hence, if m = 1 then ψ′ ⊥ ⟨−cde⟩ must be the hyperbolic plane, thus ξ′ = 0 by injectivity of s_{2} and ξ = {−ce, −de}. If m > 1 then by induction hypothesis ξ′ is a sum of m − 1 symbols, which implies that ξ is a sum of m symbols. □

Lemma 3.3. Suppose that char(F) ≠ 2. Let a ∈ F^{×}. If a can be written in F as a sum of three squares (or less), then there exists b ∈ F^{×} such that in k_{2}F we have {−1, −1} = {−a, −b}.

Proof. If a is a sum of three squares in F, then there is some b ∈ F^{×} such that ⟨⟨−1, −1⟩⟩ = ⟨1, 1, 1, 1⟩ = ⟨1, a, b, ab⟩ = ⟨⟨−a, −b⟩⟩; hence s_{2}({−1, −1}) = s_{2}({−a, −b}) and thus {−1, −1} = {−a, −b}, by injectivity of s_{2}. □

Proposition 3.4. Suppose that |F^{×}/F^{×}^{2}| = 2^{m} where m > 1 and that there is an element ξ ∈ k_{2}F which cannot be written as a sum of [\frac{m}{2}] symbols. Then there exist a_{1}, b_{1}, …, a_{n}, b_{n} ∈ F^{×}, where 2n + 1 = m, such that in k_{2}F we have

ξ = {−1, −1} + {−a_{1}, −b_{1}} + ⋯ + {−a_{n}, −b_{n}} .

Furthermore, given any such representation of ξ as a sum of n symbols, the elements {−1}, {a_{1}}, {b_{1}}, …, {a_{n}}, {b_{n}} form an 𝔽_{2}-basis of k_{1}F and none of the elements a_{1}, b_{1}, …, a_{n}, b_{n} is a sum of squares in F.

Proof. By (2.3(a)) and by the hypothesis on ξ, the element −1 cannot be a square in F, in particular, char(F) ≠ 2.

Let θ be an element of Λ2_{{−1}}(k_{1}F) which maps to ξ under the canonical homomorphism Λ2_{\{−1\}}(k_{1}F) ⟶ k_{2}F. The hypothesis implies that θ is not a sum of [\frac{m}{2}] pure 2-vectors in Λ2_{{−1}}(k_{1}F). Since k_{1}F has dimension m, we conclude by (2.7) that m is odd and that θ is a sum of \frac{m+1}{2} pure 2-vectors where the first one is \{−1\}∧\{−1\}. Hence we may choose elements a_{1}, b_{1}, …, a_{n}, b_{n} ∈ F^{×}, where 2n+1 = m, such that

θ = {−1} ∧ {−1} + {−a_{1}} ∧ {−b_{1}} + ⋯ + {−a_{n}} ∧ {−b_{n}}.

Further, using {−1} + {−x} = {x} in k_{1}F for any x ∈ F^{×}, we conclude from the second part of (2.7) that the elements {−1}, {a_{1}}, {b_{1}}, …, {a_{n}}, {b_{n}} form an 𝔽_{2}-basis of k_{1}F.

From the above representation of θ we obtain immediately that

ξ = {−1, −1} + {−a_{1}, −b_{1}} + ⋯ + {−a_{n}, −b_{n}}.

The proof will be complete if we can show for any l ∈ ℕ and any representation of ξ as above, that none of the elements a_{1}, b_{1}, …, a_{n}, b_{n} can be a sum of l squares in F. Suppose that this statement is true for a certain positive integer l. To show that it still holds after replacing l by l + 1, we may assume that a_{1} is a sum of l + 1 squares in F, in order to derive a contradiction.

Since {−1, −1} + {−a_{1}, −b_{1}} cannot be equal to a symbol, a_{1} cannot be a sum of three squares in F by Lemma 3.3, hence l ⩾ 3. By our assumption we may write a_{1} = c + e with elements c, e ∈ F^{×}, where c is a sum of two squares and e is a sum of l − 1 but not a sum of fewer squares in F.

As l − 1 ⩾ 2, the element e is not a square in F. Since there exists an element of F which is a sum of l + 1 but not of l squares in F, the form l × ⟨1⟩ over F is necessarily anisotropic. It follows that e ∉ −F^{×}^{2}. As a_{1} is not a sum of l squares in F, we also obtain that e ∉ a_{1}F^{×}^{2}. Further, by the equality a_{1} = (a_{1}^{2} − ea_{1})/c and since c is a sum of 2 squares in F while this is not the case for a_{1}, we see that −ea_{1} is not a square in F, i.e. e ∉ −a_{1}F^{×}^{2}. Until here we have shown that e ∉ ±F^{×}^{2} ∪ ±a_{1}F^{×}^{2}.

In k_{2}F we have the equality {−a_{1}, e} = {−c, a_{1}e}. Since c and e are sums of l squares in F, we get from the induction hypothesis that the symbol {−a_{1}, −b_{1}} cannot be equal neither to {−a_{1}, −e} nor to {−a_{1}, e}. Hence the symbols {−a_{1}, −eb_{1}} and {−a_{1}, eb_{1}} are both nonzero. We conclude that e ∉ ±b_{1}F^{×}^{2} ∪ ±a_{1}b_{1}F^{×}^{2}.

Since {−1}, {a_{1}}, {b_{1}}, …, {a_{n}}, {b_{n}} form an 𝔽_{2}-basis of k_{1}F, we may write e = xy where

x ∈ ±F^{×}^{2} ∪ ±a_{1}F^{×}^{2} ∪ ±b_{1}F^{×}^{2} ∪ ±a_{1}b_{1}F^{×}^{2}

and where y is equal to 1 or to a nontrivial product of some of the elements a_{2}, b_{2}, …, a_{n}, b_{n}. Since e ∉ ±F^{×}^{2} ∪ ±a_{1}F^{×}^{2} ∪ ±b_{1}F^{×}^{2} ∪ ±a_{1}b_{1}F^{×}^{2} as we have shown above, we have y ≠ 1. Therefore (2.6(b)) shows that in Λ2_{{−1}}(k_{1}F) the element {−a_{2}} ∧ {−b_{2}} + ⋯ + {−a_{n}} ∧ {−b_{n}} can be rewritten as a sum of (n − 1) pure 2-vectors where the first is of the shape {±y} ∧ {∗}. Hence we may suppose that a_{2} = ±y. By the relations in Λ2_{{−1}}(k_{1}F) and since e = xy = ±a_{2}x where x is as described above, we can rewrite {−a_{1}} ∧ {−b_{1}} + {−a_{2}} ∧ {−b_{2}} as a sum of two pure 2-vectors which is either of the shape {−a_{1}} ∧ {∗} + {±e} ∧ {∗} or of the shape {−a_{1}} ∧ {±e} + {∗} ∧ {∗}. So we may actually suppose that e is equal to one of ±b_{1} or ±a_{2}. However, by the induction hypothesis, neither b_{1} nor a_{2} can be a sum of l squares in F. Hence e is equal either to −a_{2} or to −b_{1}. In particular, since a_{1} = c+e where c is a sum of 2 squares in F, the quadratic form ⟨1, 1, −a_{1}, −b_{1}, −a_{2}⟩ is isotropic.

We put ζ = {−1, −1} + {−a_{1}, −b_{1}} + {−a_{2}, −b_{2}} ∈ k_{2}F. In Ī^{2}F we compute

s_{2}(ζ) = \overline{⟨⟨−1, −1⟩⟩} + \overline{(−1) · ⟨⟨−a_{1}, −b_{1}⟩⟩} + \overline{(−1) · ⟨⟨−a_{2}, −b_{2}⟩⟩}
= \overline{⟨1, 1, −a_{1}, −b_{1}, −a_{1}b_{1}, −a_{2}, −b_{2}, −a_{2}b_{2}⟩} .

By the above, the 8-dimensional form ⟨1, 1, −a_{1}, −b_{1}, −a_{1}b_{1}, −a_{2}, −b_{2}, −a_{2}b_{2}⟩ is isotropic, hence Witt equivalent to a 6-dimensional form φ. Then s_{2}(ζ) = \overline{φ}, thus ζ is equal to a sum of two symbols by (3.2). But then ξ can be written as a sum of n < \frac{m}{2} symbols, in contradiction to the hypothesis. □

Corollary 3.5. If F is a nonreal field such that |F^{×}/F^{×}^{2}| = 2^{m} then every element of k_{2}F can be written as a sum of [\frac{m}{2}] symbols.

Proof. For m = 1 this is clear by (3.1). Assume that m > 1 and that there exists an element in k_{2}F which cannot be written as a sum of [\frac{m}{2}] symbols. Then we must have −1 ∉ F^{×}^{2} by (2.9), and by the last proposition there exist elements in F which are not sums of squares. Therefore F is real. □

The last corollary completes the proof of Theorem 1.1 (see Remark 2.9).

If the field F is nonreal, then one denotes by s(F) the least positive integer s such that −1 is a sum of s squares over F, otherwise one puts s(F) = ∞. The invariant s(F) is called the level of F. By a famous result due to Pfister, the integers occurring as the level of some field are precisely the powers of 2. However, it is still an unsolved problem whether there is a nonreal field F of level greater than 4 which has finite square class group F^{×}/F^{×}^{2}. (For further information on this problem, see [2] and the references given there.) If the answer to this question turned out to be negative then one could simplify the proof of Proposition 3.4.

We will see in the next section that, whenever there exist fields of finite level s and with finite square class group, then for m sufficiently large there is such a field L such that, in addition to s(L) = s, one has |L^{×}/L^{×}^{2}| = 2^{m} and λ_{2}(L) = [\frac{m}{2}] (see Corollary 4.3). This shows in particular that the estimate given in Corollary 3.5 is generally the best possible.

4. Symbols and central simple algebras

Let p be a prime number. We suppose for the moment that the field F contains a primitive p-th root of unity ω; in particular, char(F) ≠ p. Given a, b ∈ F^{×} we denote by (a, b)_{F,ω} the central simple F -algebra of degree p generated by two elements α and β which are subject to the relations α^{p} = a, β^{p} = b and βα = ωαβ. We call this a symbol algebra of degree p over F. In the case p = 2, we have ω = −1 and, hence, we recover the definition of an F -quaternion algebra; we then write (a, b)_{F} instead of (a, b)_{F,−1}.

Let Br(F) denote the Brauer group of F and by Br_{p}(F) its p-torsion subgroup. There exists a canonical group homomorphism K_{2}^{(p)}F ⟶ Br_{p}(F) which maps a symbol {a, b} ∈ K_{2}^{(p)}F, with a, b ∈ F^{×}, to the class of (a, b)_{F,ω} in Br_{p}(F). By the Merkurjev–Suslin theorem (cf. [14] or [13]), this is actually an isomorphism, but we shall not use this fact in the sequel.

Lemma 4.1. Suppose that the field F contains a primitive p-th root of unity. Let a_{1}, b_{1}, …, a_{n}, b_{n} ∈ F^{×}. If (a_{1}, b_{1})_{F,ω} ⊗_{F} ⋯ ⊗_{F} (a_{n}, b_{n})_{F,ω} is a division algebra then the element {a_{1}, b_{1}} + ⋯ + {a_{n}, b_{n}} in K_{2}^{(p)}F cannot be written as a sum of less than n symbols.

Proof. We put ξ := {a_{1}, b_{1}} + ⋯ + {a_{n}, b_{n}}. Let D denote the central F -division algebra whose class in Br(F) is the image of ξ under the canonical homomorphism K_{2}^{(p)} ⟶ Br_{p}(F). If ξ is a sum of m symbols in K_{2}^{(p)}F, then the degree of D is at most p^{m}. On the other hand, if (a_{1}, b_{1})_{F,ω} ⊗_{F} ⋯⊗_{F} (a_{n}, b_{n})_{F,ω} is a division algebra, then it is F -isomorphic to D, which then must be of degree p^{n}. These facts together imply the statement. □

Theorem 4.2. Suppose that char(F) ≠ p and that |F^{×}/F^{×}^{p}| = p^{n}. Let m ⩾ 2n−1 and L = F ((X_{1})) … ((X_{m−n})). Then |L^{×}/L^{×}^{p}| = p^{m} and

λ_{p}(L) = \{ \begin{matrix}[\frac{m}{2}] & if p ≠ 2 or if F is nonreal, \\ [\frac{m+1}{2}] & if p = 2 and F is real. \end{matrix}

Proof. It is well-known that |L^{×}/L^{×}^{p}| = p^{m−n} · |F^{×}/F^{×}^{p}| = p^{n} To prove the remaining claims we may assume that m is equal either to 2n − 1 or to 2n. Indeed, if m > 2n then we replace n by n′ = [\frac{m}{2}] and F by F′ = F ((X_{1})) … ((X_{n′−n})), observing that |F′^{×}/F′^{×p}| = p^{n′} and that L is the iterated power series field in m − n′ variables over F′.

Let ω denote a primitive p-th root of unity. Recall that F (ω)/F is a field extension of degree dividing p − 1. Therefore any irreducible polynomial of degree p over F will stay irreducible over F (ω). For any a ∈ F^{×} \ F^{×}^{p}, the polynomial X^{p} − a is irreducible over F [9, Chapter VIII, Theorem 9.1.], hence also over F (ω). This shows that the canonical homomorphism F^{×}/F^{×}^{p} ⟶ F (ω)^{×}/F (ω)^{×}^{p} is injective. The hypothesis therefore implies |F (ω)^{×}/F (ω)^{×}^{p}| ⩾ p^{n}.

By Kummer theory (cf. [9, Chapter VIII, § 8], for example), we may choose elements a_{1}, …, a_{n} ∈ F^{×} such that F (ω, a_{1}^{1/p}, …, a_{n}^{1/p}) is an extension of F (ω) of degree p^{n}. Note that L(ω) = F (ω)((X_{1})) … ((X_{m−n})).

We assume now that m = 2n. By [22, Proposition 2.10], the central simple L(ω)- algebra (a_{1}, X_{1})_{L(ω),ω} ⊗_{L(ω)} ⋯⊗_{L(ω)} (a_{n}, X_{n})_{L(ω),ω} is a division algebra. Therefore, the element {a_{1}, X_{1}}+⋯+{a_{n}, X_{n}} in K_{2}^{(p)}L(ω) is not a sum of less than n symbols. It follows that in K_{2}^{(p)}L this element cannot be a sum of less than n symbols either. This together with Theorem 1.1 shows that λ_{p}(L) = n = [\frac{m}{2}] = [\frac{m+1}{2}].

Assume next that m = 2n−1 and that p ≠ 2 or F is nonreal. The same argument as before shows that the element {a_{1}, X_{1}} + ⋯ + {a_{n−1}, X_{n−1}} ∈ K_{2}^{(p)}L is not a sum of less than n − 1 = [\frac{m}{2}] symbols. This together with (1.1) yields λ_{p}(L) = [\frac{m}{2}].

In the remaining case, F is a real field, p = 2 and m = 2n − 1. We may then choose the elements a_{1}, …, a_{n} ∈ F^{×} as above in such a way that a_{1}, …, a_{n−1} become squares in some real closure E of F. The F -quaternion algebra (−1, −1)_{F} does not split over the field F (\sqrt{a_{1}}, …, \sqrt{a_{n−1}}), which is contained in E. Hence, by [22, Proposition 2.10], the product (−1, −1)_{L}⊗_{L}(a_{1}, X_{1})_{L}⊗_{L}⋯⊗_{L}(a_{n−1}, X_{n−1})_{L} is a division algebra. So, by (4.1), the element {−1, −1}+{a_{1}, X_{1}}+⋯+{a_{n−1}, X_{n−1}} in K_{2}^{(p)}L cannot be written as a sum of less than n symbols. By (1.1) we conclude that λ_{2}(L) = n = [\frac{m+1}{2}]. □

Corollary 4.3. Suppose that char(F) ≠ 2 and that |F^{×}/F^{×}^{2}| = 2^{n}. Let m ⩾ 2n−1 and L = F ((X_{1})) … ((X_{m−n})). Then |L^{×}/L^{×}^{2}| = 2^{m}, s(L) = s(F) and

λ_{2}(L) = \{ \begin{matrix}[\frac{m}{2}] & if F is nonreal, \\ [\frac{m+1}{2}] & otherwise. \end{matrix}

Proof. It is well-known that L has the same level as F; the rest of the statement is contained in the theorem. □

Examples 4.4. In each of the following situations, L is a nonreal field with

|L^{×}/L^{×}^{2}| = 2^{m} and λ_{2}(L) = [\frac{m}{2}] .

(1) L = ℂ((X_{1})) … ((X_{m})), where m ⩾ 0. The level of L is 1 in this case.

(2) L = 𝔽_{3}((X_{1})) … ((X_{m−1})), where m ⩾ 1 and where 𝔽_{3} denotes the field with three elements. In this case, s(L) = 2.

(3) L = ℚ_{2}((X_{1})) … ((X_{m−3})), where m ⩾ 5 and where ℚ_{2} is the field of dyadic numbers. On the other hand, a direct but tedious calculation shows that one has λ_{2}(F) = 1 for any field F of level 4 with |F^{×}/F^{×}^{2}| ⩽ 2^{4}. (This can also be seen from the results in [21].)

The key idea in the proof of the following statement goes back to Nakayama [17].

Proposition 4.5. Suppose that F contains a primitive p-th root of unity ω. Let L be an extension of F (X_{0}, …, X_{m}) contained in F (X_{0})((X_{1}))…((X_{m})). There is a tensor product of m symbol algebras of degree p over L which is a division algebra. In particular, one has λ_{p}(L) ⩾ m.

Proof. We choose distinct monic irreducible polynomials f_{1}, …, f_{m} ∈ F [X_{0}]. They represent 𝔽_{p}-linearly independent classes in F (X_{0})^{×}/F (X_{0})^{×}^{p}. Hence, [22, Proposition 2.10] implies that (f_{1}, X_{1})_{L,ω} ⊗_{L} ⋯ ⊗_{L} (f_{m}, X_{m})_{L,ω} is a central division algebra over L = F (X_{0})((X_{1}))…((X_{m})). It is now clear that the same holds if L is any subfield of F (X_{0})((X_{1}))⋯((X_{m})) containing F (X_{0}, …, X_{m}); in particular, λ_{p}(L) ⩾ m, by (4.1). □

In view of the inequality of the proposition, it seems natural to ask:

Question 4.6. Let m be a positive integer, p a prime, and C an algebraically closed field with char(F) ≠ p. Let F = C(X_{0}, …, X_{m}) be the rational function field in m variables over C. Do we have λ_{p}(F) = m?

Assume now that m = 1. Already here, the problem is unsolved and very interesting. Showing that λ_{p}(F) = 1 would imply, via the Merkurjev–Suslin Theorem, that any central division algebra over F = C(X_{0}, X_{1}) of exponent p is cyclic (equivalently, a symbol algebra of degree p). This is true at least for p ⩽ 3 (cf. [1]). By a very recent result of de Jong, any central division algebra of exponent p over F is of degree p (cf. [4]), but it is not presently known whether such an algebra is necessarily cyclic when p ⩾ 5.

An important and rather open problem in the study of symbol lengths is their behaviour under field extensions. First of all, it would be interesting to know whether λ_{p}(L) < λ_{p}(F) is possible for a prime p and a finite extension L/F of degree prime to p (cf. [7, Conjecture 3] for p = 2). Furthermore, no upper bound is known for λ_{p}(L) in terms of λ_{p}(F) and the degree of the (finite) extension L/F.

To finish, let us indicate what can happen with respect to the latter problem in the case p = 2. In our examples, the base field F will have finitely many square classes. A far more systematic discussion of the behaviour of the 2-symbol length λ_{2} under field extensions L/F of degree 2 and 3 can be found in [16].

Examples 4.7. (1) Let F_{0} be the quadratic closure of ℚ and L_{0}/F_{0} an extension of degree 3. For given m ⩾ 0, let F = F_{0}((X_{1})) … ((X_{m})) and L = L_{0}((X_{1})) … ((X_{m})). Then one has [L : F] = [L_{0} : F_{0}] = 3 and it turns out that λ_{2}(F) = [\frac{m}{2}], by (4.2), whereas λ_{2}(L) = m.

As for the last claim, one may argue as follows. By [8, Chapter VII, Appendix, Corollary 3], L_{0} has infinitely many square classes. Let a_{1}, …, a_{m} ∈ L_{0}^{×} be 𝔽_{2}- independent in L_{0}^{×}/L_{0}^{×}^{2}. Then [22, Proposition 2.10] shows that the L-algebra (a_{1}, X_{1})_{L} ⊗_{L} ⋯ ⊗_{L} (a_{n}, X_{n})_{L} is a division algebra. Using (4.1), it follows that λ_{2}(L) ⩾ m. On the other hand, a straightforward computation shows that any sum of m + 1 symbols in k_{2}L can be written as a sum of an element defined over L_{0} plus m symbols of the shape {∗, X_{i}}, 1 ⩽ i ⩽ m. However, k_{1}F_{0} = 0 and [L_{0} : F_{0}] = 3 imply that k_{2}L_{0} = 0, by [12, Lemma 2], because k_{1}F_{0} = 0 and [L_{0} : F_{0}] = 3. One conludes that λ_{2}(L) = m.

(2) Let F_{0} be a nonreal field such that |F_{0}^{×}/F_{0}^{×}^{2}| = 2^{m} for given m ⩾ 1, and such that k_{2}F_{0} = 0. Let L_{0}/F_{0} be a quadratic extension. Then it follows from [8, Chapter VII, Theorem 3.4.] that |L_{0}^{×}/L_{0}^{×}^{2}| = 2^{2m−1}. We put F = F_{0}((X_{1})) … ((X_{n−m})) and L = L_{0}((X_{1})) … ((X_{n−m})) for given n ⩾ 3m − 1. Then [L : F] = [L_{0} : F_{0}] = 2, and (4.2) yields λ_{2}(F) = [\frac{n}{2}] and λ_{2}(L) = [\frac{n+m−1}{2}]. In particular, for n = 3m − 1 we get λ_{2}(F) = [\frac{3m−1}{2}] and λ_{2}(L) = 2m − 1, so λ_{2}(L) = \frac{4λ_{2}(F)+(−1)^{m}}{3} .

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Karim Johannes Becher becher@maths.ucd.ie

Universität Konstanz Fachbereich Mathematik und Statistik, D204 78457 Konstanz Germany

Detlev W. Hoffmann detlev.hoffmann@nottingham.ac.uk

School of Mathematical Sciences University of Nottingham University Park Nottingham NG7 2RD UK