ON THE SECOND K-GROUP OF A RATIONAL FUNCTION FIELD

KARIM JOHANNES BECHER AND MÉLANIE RACZEK

Abstract. We give an optimal bound on the minimal length of a sum of symbols in the second Milnor K-group of a rational function field in terms of the degree of the ramification. Keywords: Milnor K-theory, field extension, valuation, ramification Classification (MSC 2010): 12E30, 12G05, 12Y05, 19D45

1. Introduction

Let E be an arbitrary field and F the function field of the projective line ℙ^{1} over E. For m ∈ ℕ, there is a well-known exact sequence

0 ⟶ K_{2}^{(m)}E ⟶ K_{2}^{(m)}F ⟶^{∂} ⨁_{x∈ℙ_{0}^{1}} K_{1}^{(m)}E(x)⟶K_{1}^{(m)}E ⟶ 0 , (1.1)

due to Milnor and Tate (cf. [6, (2.3)]). Here, K_{1}^{(m)} and K_{2}^{(m)} are the functors that associate to a field its first and second K-groups modulo m, respectively, and ℙ_{0}^{1} is the set of closed points of ℙ^{1}. The map ∂ is called the ramification map. By [3, (7.5.4)], for m prime to the characteristic of E, the sequence (1.1) translates into a sequence in Galois cohomology, and the proof of its exactness essentially goes back to Faddeev [2].

In this article we study how for a given element ρ in the image of ∂ one finds a good ξ ∈ K_{2}^{(m)}F with ∂(ξ) = ρ. Our main result (3.10) states that there is such a ξ that is a sum of r symbols (canonical generators of K_{2}^{(m)}F) where r is bounded by half the degree of the support of ρ. This generalizes results from [4], [7], and [8], where the problem has been studied in terms of Brauer groups in presence of a primitive mth root of unity in E for m > 0. Developing further an idea in [8, Prop. 2], we provide examples (4.3) where the bound on r cannot be improved.

2. Milnor K-theory of a rational function field

We recall the basic terminology of the K-theory for fields as introduced by Milnor [6], with slightly different notation. Let F be a field. For m, n ∈ ℕ, let K_{n}^{(m)}F denote the abelian group generated by elements called symbols, which are of the form {a_{1}, …, a_{n}} with a_{1}, …, a_{n} ∈ F^{×}, subject to the defining relations that {∗, …, ∗} : (F^{×})^{n} ⟶ K_{n}^{(m)}F is a multilinear map, that {a_{1}, …, a_{n}} = 0 whenever a_{i} + a_{i+1} = 1 in F for some i < n, and that m · {a_{1}, …, a_{n}} = 0. For a, b ∈ F^{×} we have {ab} = {a} + {b} in K_{1}^{(m)}F. The second relation above is void when n = 1, hence K_{1}^{(m)}F is the same as F^{×}/F^{×}^{m}, only with different notation for the elements and the group operation. As shown in [6, (1.1) and (1.3)], it follows from the defining relations that, for a_{1}, …, a_{n} ∈ F^{×}, we have {a_{σ(1)}, …, a_{σ(n)}} = ε{a_{1}, …, a_{n}} for any permutation σ of the numbers 1, …, n with signature ε = ±1, and furthermore {a_{1}, …, a_{n}} = 0 whenever a_{i} + a_{i+1} = 0 for some i < n.

Date: February 21, 2012.

With these notations, K_{n}^{(0)}F is the full Milnor K-group K_{n}F introduced in [6], and K_{n}^{(m)}F is its quotient modulo m for m ≥ 1.

By a ℤ-valuation we mean a valuation with value group ℤ. Given a ℤ-valuation v on F we denote by O_{v} its valuation ring and by κ_{v} its residue field. For a ∈ O_{v} let \overline{a} denote the natural image of a in κ_{v}. By [6, (2.1)], for n ≥ 2 and a ℤvaluation v on F, there is a unique homomorphism ∂_{v} : K_{n}^{(m)}F ⟶ K_{n−1}^{(m)}κ_{v} such that

∂_{v}(\{f, g_{2}, …, g_{n}\}) = v(f) · \{\overline{g}_{2}, …, \overline{g_{n}}\}

for f ∈ F^{×} and g_{2}, …, g_{n} ∈ O_{v}^{×}. When n = 2, for f, g ∈ F^{×} we have f^{−v(g)}g^{v(f)} ∈ O_{v}^{×} and

∂_{v}(\{f, g\}) = \{(−1)^{v(f)v(g)}\overline{f^{−v(g)}g^{v(f)}}\} in K_{1}^{(m)}κ_{v} .

We turn to the situation where F is the function field of ℙ^{1} over E. By the choice of a generator, we identify F with the rational function field E(t) in the variable t over E. Let P denote the set of monic irreducible polynomials in E[t]. Any p ∈ P determines a ℤ-valuation v_{p} on E(t) that is trivial on E and such that v_{p}(p) = 1. There is further a unique ℤ-valuation v_{∞} on E(t) such that v_{∞}(f) = − deg(f) for any f ∈ E[t] \ {0}. We set P′ = P ∪ {∞}. For p ∈ P′ we write ∂_{p} for ∂_{v_{p}} and we denote by E_{p} the residue field of v_{p}. Note that E_{p} is naturally isomorphic to E[t]/(p) for p ∈ P, and E_{∞} is naturally isomorphic to E.

It follows from [6, Sect. 2] that the sequence

(2.1)  0 ⟶ K_{n}^{(m)}E ⟶ K_{n}^{(m)}E(t) \underrightarrow{⨁ ∂_{p}} ⨁_{p∈P} K_{n−1}^{(m)}E_{p} ⟶ 0

is split exact. We are going to reformulate this fact for n = 2 and to relate (2.1) to (1.1). We set

ℜ′_{m}(E) = ⨁_{p∈P′} K_{1}^{(m)}E_{p} .

For p ∈ P′, the norm map of the finite extension E_{p}/E yields a group homomorphism K_{1}^{(m)}E_{p} ⟶ K_{1}^{(m)}E. Summation over these maps for all p ∈ P′ yields a homomorphism N : ℜ′_{m}(E) ⟶ K_{1}^{(m)}E . Let ℜ_{m}(E) denote the kernel of N. We set ∂ = ⨁_{p∈P′} ∂_{p}. By [3, (7.2.4) and (7.2.5)] we obtain an exact sequence

0 ⟶ K_{2}^{(m)}E ⟶ K_{2}^{(m)}E(t) ⟶^{∂} ℜ′_{m}(E) ⟶^{N} K_{1}^{(m)}E ⟶ 0 . (2.2)

In particular, ℜ_{m}(E) is equal to the image of ∂ : K_{2}^{(m)}E(t) ⟶ ℜ′_{m}(E).

The choice of the generator of F over E fixes a bijection ϕ : ℙ_{0}^{1} ⟶ P′ and for x ∈ ℙ_{0}^{1} a natural isomorphism between E(x) and E_{ϕ(x)}. This identifies ⨁_{x∈ℙ_{0}^{1}} K_{1}^{(m)}E(x) with ℜ′_{m}(E), and further the sequence (1.1) with (2.2). We will work with (2.2) in the sequel.

For ρ = (ρ_{p})_{p∈P′} ∈ ℜ′_{m}(E) we denote Supp(ρ) = {p ∈ P′ | ρ_{p} ≠ 0} and deg(ρ) = ∑_{p∈Supp(ρ)}[E_{p} : E], and call this the support and the degree of ρ. The degree of an element of ℜ′_{m}(E) is invariant under automorphisms of E(t)/E.

3. Bound for representation by symbols in terms of the degree

In this section we study the relation between the degree of ρ ∈ ℜ_{m}(E) to the properties of elements ξ ∈ K_{2}^{(m)}E(t) with ∂(ξ) = ρ. In (3.10) we will show that there always exists such ξ that is a sum of r symbols where r is the integral part of \frac{deg(ρ)}{2}. In particular, any ramification of degree at most three is realized by a symbol. This settles a question in [4, (2.5)]. In some of the following statements, we consider elements of ℜ′_{m}(E), rather than only of ℜ_{m}(E).

3.1. Proposition. If ρ ∈ ℜ_{m}(E) then deg(ρ) ≠ 1.

Proof. Consider an element ρ ∈ ℜ′_{m}(E) with deg(ρ) = 1. The support of ρ consists of one rational point p ∈ P′. Hence N(ρ) = ρ_{p} ≠ 0 in K_{1}^{(m)}E. As N ∘ ∂ = 0 it follows that ρ ∉ ℜ_{m}(E). □

We say that p ∈ P′ is rational if [E_{p} : E] = 1. We call a subset of P′ rational if all its elements are rational. We give two examples showing how to realize a given ramification of small degree and with rational support by one symbol.

3.2. Examples. (1) Let a, c ∈ E^{×} and c ∉ E^{×}^{m}. The symbol σ = {t − a, c} in K_{2}^{(m)}E(t) satisfies Supp(σ) = {t − a, ∞}, ∂_{t−a}(σ) = {c} and ∂_{∞}(σ) = {c^{−1}}.

(2) For a_{1}, a_{2}, c_{1}, c_{2} ∈ E^{×} with a_{1} ≠ a_{2}, we compute the ramification of the symbol σ = \{\frac{t−a_{1}}{c_{2}(a_{2}−a_{1})}, \frac{c_{1}(t−a_{2})}{a_{1}−a_{2}}\} in K_{2}^{(m)}E(t). It has Supp(σ) ⊆ \{t−a_{1}, t−a_{2}, ∞\}, ∂_{t−a_{i}}(σ) = {c_{i}} for i = 1, 2, and ∂_{∞}(σ) = {(c_{1}c_{2})^{−1}}.

A ramification of degree two can under some extra condition be realized by a symbol one of whose entries is a constant.

3.3. Proposition. Let ρ ∈ ℜ_{m}(E) be such that deg(ρ) = 2. If Supp(ρ) is rational or char(E) ≠ m = 2, there exist e ∈ E^{×} and f ∈ E(t)^{×} such that ρ = ∂({e, f}).

Proof. Suppose first that the support of ρ is rational. We choose a, e ∈ E^{×} such that t − a ∈ Supp(ρ) and ρ_{t−a} = {e} in K_{1}^{(m)}E. Then Supp(ρ) = {t − a, p} where p ∈ P′ is rational. As N(ρ) = 0 we obtain that ρ_{p} = {e^{−1}} in K_{1}^{(m)}E_{p}. If p = ∞, we set f = \frac{1}{t−a}. Otherwise p = t − b for some for b ∈ E, and we set f = \frac{t−b}{t−a}. In either case we obtain that ρ = ∂({e, f}).

It remains to consider the case where char(E) ≠ m = 2 and Supp(ρ) = {p} for a quadratic polynomial p ∈ P. Then E_{p}/E is a separable quadratic extension. Let x ∈ E_{p}^{×} be such that ρ_{p} = {x}. As Supp(ρ) = {p} and N(ρ) = 0, we obtain that the norm of x with respect to the extension E_{p}/E lies in E^{×2}, and therefore xE_{p}^{×2} = eE_{p}^{×2} for some e ∈ E^{×} (cf. [5, Chap. VII, (3.9)]). Hence, ρ_{p} = {x} = {e} in K_{1}^{(2)}E_{p}, and we obtain that ρ = ∂({e, p}). □

In (3.3) the rationality of the support when m ≠ 2 is not a superfluous condition; the following example was pointed out to us by J.-P. Tignol.

3.4. Example. Let k be a field. We consider the rational function field in two variables u and v over k. Let τ denote the k-automorphism of k(u, v) satisfying τ(u) = v and τ(v) = u. Then τ^{2} is the identity map on k(u, v), and E = {x ∈ k(u, v) | τ(x) = x} is a subfield of k(u, v) such that [k(u, v) : E] = 2. Consider the element y = \frac{v}{u} ∈ k(u, v). Since y ∉ E, the quadratic polynomial p = (t − y)(t − τ(y)) = t^{2} − \frac{u^{2}+v^{2}}{uv} t + 1 is irreducible over E.

Let m be an odd positive integer. We consider the symbol σ = {p, t} in K_{2}^{(m)}E(t). Note that the support of ∂(σ) is contained in \{p\} and ∂_{p}(σ) = \{\overline{t}\}. Moreover, mapping t to y induces an E-isomorphism E_{p} ⟶ k(u, v). Since y is not an mth power in k(u, v), it follows that ∂_{p}(σ) ≠ 0. Hence, Supp(∂(σ)) = {p} and deg(∂(σ)) = 2.

We claim that ∂(σ) ≠ ∂({e, f}) for any e ∈ E^{×} and f ∈ E(t)^{×}. Suppose on the contrary that there exist e ∈ E^{×} and f ∈ E(t)^{×} such that ∂_{p}(σ) = ∂_{p}({e, f}). Then we obtain that e^{v_{p}(f)}y is an mth power in k(u, v), and taking norms with respect to the extension k(u, v)/E yields that e^{2v_{p}(f)} ∈ E^{×}^{m}. Since m is odd, it follows that e^{v_{p}(f)} ∈ E^{×}^{m}, and thus ∂_{p}({e, f}) = 0, a contradiction.

The remainder of this section builds up to our main result (3.10).

3.5. Lemma. Let ρ ∈ ℜ′_{m}(E) with deg(ρ) ≥ 2. There exists a symbol σ in K_{2}^{(m)}E(t) such that deg(ρ − ∂(σ)) ≤ deg(ρ) − 1 and such that this inequality is strict except possibly when ∂_{∞}(σ) ≠ 0 = ρ_{∞} or deg(ρ) = 2. More precisely, one may choose σ = {fh, g} where f is the product of the polynomials in Supp(ρ) and g, h ∈ E[t] \ {0} are such that deg(g) < deg(f) and, either deg(h) < deg(g), or gh ∈ E^{×}.

Proof. Let f be the product of the polynomials in Supp(ρ). By the Chinese Remainder Theorem, we may choose g ∈ E[t] prime to f with deg(g) < deg(f) such that ∂_{p}({f, g}) = ρ_{p} for all monic irreducible polynomials p ∈ Supp(ρ). If g is constant, let h = 1. If g is not square-free, let h be the product of the different monic irreducible factors of g. If g is square-free and not constant, then using the Chinese Remainder Theorem we choose h ∈ E[t] prime to g with deg(h) < deg(g) such that ∂_{p}(\{f, g\}) − ρ_{p} = \{\overline{h}\} in K_{1}^{(m)}E_{p} for every monic irreducible factor p of g. Then g, h and σ = {fh, g} have the desired properties. □

3.6. Lemma. Let d ∈ ℕ \ {0} and f ∈ E[t] non-constant and square-free such that deg(p) ≥ d for every irreducible factor p of f. Let F = E[t]/(f) and let ϑ denote the class of t in F. For any a ∈ F^{×} there exist nonzero polynomials g, h ∈ E[t] with deg(h) ≤ d − 1 and deg(g) ≤ deg(f) − d such that a = \frac{g(ϑ)}{h(ϑ)}.

Proof. Let V = ⨁_{i=0}^{d−1} Eϑ^{i} and W = ⨁_{i=0}^{e−d} Eϑ^{i} where e = deg(f). By the choice of d and the Chinese Remainder Theorem, we have V \ {0} ⊆ F^{×}, where F^{×} denotes the group of invertible elements of F. As a ∈ F^{×} we have dim_{E}(V a) = dim_{E}(V) = d and dim_{E}(V a) + dim_{E}(W) = e + 1 > e = [F : E], so V a ∩ W ≠ 0. Therefore h(ϑ)a = g(ϑ) for certain h, g ∈ E[t] \ {0} with deg(h) ≤ d − 1 and deg(g) ≤ e − d. Thus h(ϑ) ∈ V \ \{0\} ⊆ F^{×} and a = \frac{g(ϑ)}{h(ϑ)}. □

3.7. Lemma. Let ρ ∈ ℜ′_{m}(E) and q ∈ Supp(ρ) such that deg(q) = 2n + 1 with n ≥ 1. There exists a symbol σ in K_{2}^{(m)}E(t) such that deg(ρ−∂(σ)) ≤ deg(ρ)−2. More precisely, one may choose σ = {qhf^{−2}g^{−2}, g^{−1}f} with f, g, h ∈ E[t] \ {0} such that deg(f), deg(g) ≤ n and deg(h) ≤ 2n − 1.

Proof. Using (3.6) we choose f, g ∈ E[t] \ {0} with deg(f), deg(g) ≤ n such that ∂_{q}({q, g^{−1}f}) = ρ_{q}. Then q is prime to fg. If fg is constant, let h = 1. If fg is not square-free, let h be the product of the different monic irreducible factors of fg. If fg is square-free and not constant, we choose h ∈ E[t] prime to fg and with deg(h) < deg(fg) such that ∂_{p}({h, g^{−1}f}) = ∂_{p}({q^{−1}f^{2}g^{2}, g^{−1}f}) for every monic irreducible factor p of fg. In any case deg(h) ≤ 2n − 1 = deg(q) − 2.

Let σ = {qhf^{−2}g^{−2}, g^{−1}f}. We have ∂_{q}(σ) = ρ_{q} and ∂_{p}(σ) = 0 for every monic irreducible polynomial p ∈ E[t] prime to h and not contained in Supp(ρ). It follows that q ∈ Supp(ρ)\Supp(ρ−∂(σ)) and that every polynomial in Supp(ρ− ∂(σ)) \ Supp(ρ) divides h. Furthermore, if deg(h) = 2n − 1, then deg(f) = deg(g) = n, so that deg(qh) = 4n = 2 deg(fg) and thus ∂_{∞}(σ) = 0. We conclude that deg(ρ − ∂(σ)) ≤ deg(ρ) − 2 in any case. □

3.8. Proposition. Let ρ ∈ ℜ′_{m}(E) with deg(ρ) ≥ 2. There exists a symbol σ in K_{2}^{(m)}E(t) such that deg(ρ − ∂(σ)) ≤ deg(ρ) − 1. Moreover, if deg(ρ) ≥ 3 and Supp(ρ) contains an element of odd degree, then there exists a symbol σ in K_{2}^{(m)}E(t) such that deg(ρ − ∂(σ)) ≤ deg(ρ) − 2.

Proof. In view of (3.5) only the second part of the statement remains to be proven. If Supp(ρ) contains a non-rational point of odd degree, the statement follows from (3.7). Suppose now that Supp(ρ) contains a rational point. Note that the statement is invariant under E-automorphisms of E(t). Hence, we may assume that ∞ ∈ Supp(ρ), in which case the statement follows from (3.5). □

3.9. Question. Given ρ ∈ ℜ_{m}(E) with deg(ρ) ≥ 3, does there always exist a symbol σ in K_{2}^{(m)}E(t) such that deg(ρ − ∂(σ)) ≤ deg(ρ) − 2?

For x ∈ ℝ, the unique z ∈ ℤ such that z ≤ x < z + 1 is denoted ⌊x⌋.

3.10. Theorem. For ρ ∈ ℜ_{m}(E) and n = ⌊\frac{deg(ρ)}{2}⌋, there exist symbols σ_{1}, …, σ_{n} in K_{2}^{(m)}E(t) such that ρ = ∂(σ_{1} + ⋯ + σ_{n}).

Proof. We proceed by induction on n. If n = 0 then ρ = 0 by (3.1) and the statement is trivial. Assume that n > 0. We have either deg(ρ) = 2n+1, in which case ρ contains a point of odd degree, or deg(ρ) = 2n. Hence, by (3.8) there exists a symbol σ in K_{2}^{(m)}E(t) with deg(ρ−∂(σ)) ≤ 2n−1. By the induction hypothesis there exist symbols σ_{1}, …, σ_{n−1} in K_{2}^{(m)}E(t) with ρ − ∂(σ) = ∂(σ_{1} + ⋯ + σ_{n−1}). Then ρ = ∂(σ_{1} + ⋯ + σ_{n−1} + σ). □

If we knew that for m ≥ 1 every element of ℜ_{m}(E) had a lift to ℜ_{0}(E) of the same degree, it would be sufficient to formulate and prove (3.10) for m = 0.

4. Example showing that the bound is sharp

In this section we show that the bound (3.10) is sharp for all m and in all degrees. In order to obtain an example in (4.3) where the bound of (3.10) is an equality, we adapt Sivatski’s argument in [8, Prop. 2].

For any a ∈ E, there is a unique homomorphism s_{a} : K_{n}^{(m)}E(t) ⟶ K_{n}^{(m)}E such that s_{a}({f_{1}, …, f_{n}}) = {f_{1}(a), …, f_{n}(a)} for any f_{1}, …, f_{n} ∈ E[t] prime to t − a and such that s_{a}({t − a, ∗, …, ∗}) = 0 (cf. [3, (7.1.4)]).

4.1. Lemma. The homomorphism s = s_{0} − s_{1} : K_{n}^{(m)}E(t) ⟶ K_{n}^{(m)}E has the following properties:

(a) s(K_{n}^{(m)}E) = 0,

(b) s({(1 − a)t + a, b_{2}, …, b_{n}}) = {a, b_{2}, …, b_{n}} for any a, b_{2}, …, b_{n} ∈ E^{×},

(c) any symbol in K_{n}^{(m)}E(t) is mapped under s to a sum of two symbols in K_{n}^{(m)}E.

Proof. Since s_{0} and s_{1} both restrict to the identity on K_{n}^{(m)}E, part (a) is clear. For a, b_{2}, …, b_{n} ∈ E^{×} and σ = {(1 − a)t + a, b_{2}, …, b_{n}}, we have s_{1}(σ) = 0 and thus s(σ) = s_{0}(σ) = {a, b_{2}, …, b_{n}}. This shows (b). Part (c) follows from the observation that both s_{0} and s_{1} map symbols to symbols. □

4.2. Proposition. Let d ∈ ℕ, a_{1}, …, a_{d} ∈ E^{×}, and σ_{1}, …, σ_{d} symbols in K_{n−1}^{(m)}E. Assume that ∑_{i=1}^{d}{a_{i}} · σ_{i} ∈ K_{n}^{(m)}E is not equal to a sum of less than d symbols and let

ξ = ∑_{i=1}^{d} {(1 − a_{i})t + a_{i}} · σ_{i} ∈ K_{n}^{(m)}E(t) .

Then deg(∂(ξ)) = d + 1, and if r ∈ ℕ is such that ∂(ξ) = ∂(τ_{1} + ⋯ + τ_{r}) for symbols τ_{1}, …, τ_{r} in K_{n}^{(m)}E(t), then r ≥ ⌊\frac{d+1}{2}⌋.

Proof. The hypothesis that ξ cannot be written as a sum of less than d symbols has a few consequences. For i = 1, …, d, it follows that {a_{i}} · σ_{i} ≠ 0, so in particular a_{i} ≠ 1, and with p = t − \frac{a_{i}}{1−a_{i}} we get that ∂_{p}(ξ) = σ_{i} ≠ 0 in K_{n−1}^{(m)}E. Furthermore, we obtain that ∂_{∞}(ξ) = − ∑_{i=1}^{d} σ_{i} ≠ 0 in K_{n−1}^{(m)}E. Therefore we have Supp(∂(ξ)) = \{t − \frac{a_{i}}{1−a_{i}} | 1 ≤ i ≤ d\} ∪ \{∞\} and thus deg(∂(ξ)) = d + 1.

Assume now that r ∈ ℕ and ∂(ξ) = ∂(τ_{1} + ⋯ + τ_{r}) for symbols τ_{1}, …, τ_{r} in K_{n}^{(m)}E(t). Then τ_{1} + ⋯ + τ_{r} − ξ is defined over E. Let s be the map from (4.1). By (4.1) we obtain that s(τ_{1} + ⋯ + τ_{r} − ξ) = 0 and thus

∑_{i=1}^{d}{a_{i}} · σ_{i} = s(ξ) = s(τ_{1}) + … + s(τ_{r}) ∈ K_{n}^{(m)}E ,

which is a sum of 2r symbols. Hence 2r ≥ d, by the hypothesis on d. □

4.3. Example. Let p be a prime dividing m. Let k be a field containing a primitive pth root of unity ω and a_{1}, …, a_{d} ∈ k^{×} such that the Kummer extension k(\sqrt[p]{a_{1}}, …, \sqrt[p]{a_{d}}) of k has degree p^{d}. Let b_{1}, …, b_{d} be indeterminates over k and set E = k(b_{1}, …, b_{d}). Using [9, (2.10)] and [1, (2.1)], it follows that ∑_{i=1}^{d}{a_{i}, b_{i}} is not equal to a sum of less than d symbols in K_{2}^{(p)}E. Since p divides m, it follows immediately that ∑_{i=1}^{d}{a_{i}, b_{i}} ∈ K_{2}^{(m)}E is not a sum of less than d symbols in K_{2}^{(m)}E. Consider ξ = ∑_{i=1}^{d}{(1 − a_{i})t + a_{i}, b_{i}} in K_{2}^{(m)}E(t). By (4.2), for ρ = ∂(ξ) we have that deg(ρ) = d + 1 and ρ ≠ ∂(ξ′) for any ξ′ ∈ K_{2}^{(m)}E(t) that is a sum of less than r = ⌊\frac{deg(ρ)}{2}⌋ symbols.

Acknowledgements. We wish to express our gratitude to Jean-Pierre Tignol for his interest in our work and all his support in its course. This work was done while the first named author was a Fellow of the Zukunftskolleg and the second named author was a Postdoctoral Fellow of the Fonds de la Recherche Scientifique – FNRS. The project was further supported by the Deutsche Forschungsgemeinschaft (project Quadratic Forms and Invariants, BE 2614/3).

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Universität Konstanz, Zukunftskolleg / FB Mathematik und Statistik, 78457 Konstanz, Germany.

  E-mail address: becher@maths.ucd.ie

Université catholique de Louvain, ICTEAM, Chemin du Cyclotron 2, 1348 Louvain-La-Neuve, Belgium.

  E-mail address: melanie.raczek@ulouvain.be