RAMIFICATION SEQUENCES AND BEZOUTIAN FORMS

KARIM JOHANNES BECHER AND MÉLANIE RACZEK

Abstract. We continue our study from [1] on the problem to bound the number of symbols needed to obtain an element of the second K-group of a rational function field with given ramification. Here we focus on the case of Milnor K-groups modulo 2 for fields of characteristic different from 2. To a given ramification sequence, we associate a quadratic form defined over the base field and study its properties. In particular, we relate the Witt index of the quadratic form to the minimal number of symbols necessary to represent the ramification sequence. Keywords: Milnor K-theory, quadratic form, valuation, ramification Classification (MSC 2010): 12E30, 12G05, 12Y05, 19D45

1. Introduction

Let K_{1} and K_{2} denote the functors associating to a field its first and second K-group. Let E be a field and let F be the function field of the projective line ℙ_{E}^{1} over E. Let ℙ_{E}^{1(1)} denote the set of closed points of ℙ_{E}^{1}.

Let m ∈ ℕ. There is a natural exact sequence

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

due to Tate (cf. [7, (2.3)]), where ∂ denotes the ramification map. By a ramification sequence we mean an element in the direct sum term whose image in K_{1}E/m is trivial and which therefore lies in the image of ∂. The group K_{2}F/m has a canonical set of generators, which are called symbols. Given a ramification sequence ρ one may ask how many symbols are needed to obtain an element ξ ∈ K_{2}F/m such that ∂(ξ) = ρ. Without restrictions on E, in [1] for a ramification sequence ρ we constructed an element ξ ∈ K_{2}F/m such that ∂(ξ) = ρ and ξ is a sum of r symbols, where r is the integral part of the degree of ρ divided by two. We further gave examples where this bound is best possible.

We continue our investigation of the problem of bounding the number of symbols needed for representing a given ramification, but restricted to the case where m = 2 and char(E) ≠ 2. Thus we consider the K-group k_{2}F = K_{2}F/2 which by Merkurjev’s Theorem is naturally isomorphic to Br_{2}(F), the 2-torsion part of the Brauer group of F. Hence, the problem can be reinterpreted as a problem in terms of central simple algebras, where symbols correspond to quaternion algebras.

Date: October 20, 2016.

In this setting we relate the problem to quadratic form theory. The connection is made via pairs (f, g) of square-free coprime polynomials f, g ∈ E[t]. On the one hand, any ramification sequence can be represented by such a pair. On the other hand, any such pair gives rise to a quadratic form over E, closely related to the so-called Bezoutian of f and g introduced by Sylvester and Cayley.

In Section 2 we recall the necessary notions and statements from Milnor’s K-theory involved in the above exact sequence (in the case of k_{2}). In Section 3 we study the properties of the Bezoutian of a pair of polynomials (f, g) as above in our setting. In particular, we characterise the isotropy of this form (Proposition 3.4) and we obtain a reciprocity law (Corollary 3.9), which together will be crucial in Section 5.

In Section 4 we consider a pair (f, g) coming from a specific representation of a symbol and show that the associated Bezoutian is hyperbolic provided that the symbol is unramified at ∞ (Proposition 4.4).

In Section 5, given a ramification sequence ρ we construct a sequence of polynomials which yields a representation of ρ as a sum of symbols. If ρ is given by the pair (f, g), we obtain an upper bound on the number of symbols needed to represent ρ in terms of the Witt index of the Bezoutian of (f, g) (Theorem 5.9). It refines the bound from [1, Theorem 3.10]: the larger the Witt index, the better the improvement.

The final section illustrates our study by results and examples for ramification sequences of small degree. In particular, we obtain a criterion for a ramification sequence of degree four to be represented by a symbol (Theorem 6.1). This generalises a result due to Sivatski in [8, Prop. 1.4], which inspired our investigation.

2. Preliminaries

Let F be a field. Let k_{2}F denote the abelian group generated by symbols, which are elements of the form {a, b} with a, b ∈ F^{×}, subject to the defining relations that the pairing

{·, ·} : F^{×} × F^{×} ⟶ k_{2}F

is bilinear, {a, 1 − a} = 0 for any a ∈ F^{×} \ {1} and {a^{2}, b} = 0 for all a, b ∈ F^{×}. It follows from the defining relations that k_{2}F is 2-torsion and that, for any a, b ∈ F^{×}, we have {a, b} = {b, a}, and further {a, b} = {a + b, −ab} provided that b ≠ −a. Let k_{1}F denote the square class group F^{×}/F^{×2} in additive notation, where we write {a} ∈ k_{1}F for the element given by the square class of a ∈ F^{×} and thus have {a}+{b} = {ab} for a, b ∈ F^{×}. Note that k_{i}F = K_{i}F/2 for i = 1, 2 in the notation used in the introduction.

By a global field we mean a field F that is either a finite extension of ℚ or a function field in one variable over a finite field.

2.1. Proposition. If F is a global field then every element of k_{2}F is a symbol.

Proof. This follows from the corresponding statement for K_{2}F. It was shown in [6] that every element of K_{2}F is a symbol, and as k_{2}F = K_{2}F/2 our statement follows. If char(F) ≠ 2 then one can argue alternatively that by the Hasse-Minkowski Theorem any quadratic form in six variables of determinant −1 is isotropic, which implies the statement. □

By a ℤ-valuation on a field 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 [7, (2.1)], for n ⩾ 2 and a ℤ-valuation v on F, there is a unique homomorphism ∂_{v} : k_{2}F ⟶ k_{1}κ_{v} such that

∂_{v}(\{f, g\}) = v(f) · \{\overline{g}\} in k_{1}κ_{v}

for f ∈ F^{×} and g ∈ O_{v}^{×}. For f, g ∈ F^{×} we obtain that 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}κ_{v} .

We turn to the situation where F is the function field of ℙ_{E}^{1} over a field 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, for p ∈ P, E_{p} is naturally isomorphic to E[t]/(p), and E_{∞} is naturally isomorphic to E.

It follows from [7, Sect. 2] that the sequence

(2.2)  0 ⟶ k_{2}E ⟶ k_{2}E(t) \underrightarrow{⨁ ∂_{p}} ⨁_{p∈P} k_{1}E_{p} ⟶ 0

is split exact. We reformulate this fact and relate (2.2) to (1.1). We set

ℜ′_{2}(E) = ⨁_{p∈P′} k_{1}E_{p} .

(The index 2 is a reminiscence to the fact that we are working with Milnor Kgroups modulo 2.)

For p ∈ P′, the norm map of the finite extension E_{p}/E yields a group homomorphism k_{1}E_{p} ⟶ k_{1}E. Summation over these maps for all p ∈ P′ yields a homomorphism N : ℜ′_{2}(E) ⟶ k_{1}E . Let ℜ_{2}(E) denote the kernel of N. We call ∂ = ⨁_{p∈P′} ∂_{p} the ramification map. By [2, (7.2.4) and (7.2.5)] we obtain an exact sequence

0 ⟶ k_{2}E ⟶ k_{2}E(t) ⟶^{∂} ℜ′_{2}(E) ⟶^{N} k_{1}E ⟶ 0 . (2.3)

In particular, ℜ_{2}(E) is equal to the image of ∂ : k_{2}E(t) ⟶ ℜ′_{2}(E). The elements of ℜ_{2}(E) are therefore called ramification sequences.

The choice of the generator of F over E fixes a bijection ϕ : ℙ_{E}^{1(1)} ⟶ P′ and for any x ∈ ℙ_{E}^{1(1)} a natural isomorphism between E(x) and E_{ϕ(x)}. This identifies ⨁_{x∈ℙ_{E}^{1(1)}} k_{1}E(x) with ℜ′_{2}(E), and further the sequence (1.1) with (2.3). We will work with (2.3) in the sequel.

For a finite set S ⊆ P′ we call ∑_{p∈S}[E_{p} : E] the degree of S and denote it by deg(S). For ρ = (ρ_{p})_{p∈P′} ∈ ℜ′_{2}(E) we denote Supp(ρ) = {p ∈ P′ | ρ_{p} ≠ 0} and deg(ρ) = deg(Supp(ρ)), and we call this the support and the degree of ρ.

Given a ramification sequence ρ ∈ ℜ_{2}(E), we say that ρ is represented by ξ ∈ k_{2}E(t) if ∂(ξ) = ρ. The problem at the centre of our study is to obtain a good upper bound on the smallest n ∈ ℕ such that ρ is represented by a sum of n symbols in k_{2}E(t).

2.4. Example. Assume that the field E is finite. Then by Proposition 2.1 every element of k_{2}E(t) is a symbol. In particular, any ramification sequence in ℜ_{2}(E) is represented by a symbol.

For x ∈ ℝ we set ⌊x⌋ = max{z ∈ ℤ | z ⩽ x} and ⌈x⌉ = min{z ∈ ℤ | z ⩾ x}.

In [1, Theorem 3.10] we proved the following statement:

2.5. Theorem. Any ramification sequence ρ ∈ ℜ_{2}(E) is represented by a sum of n symbols for n = ⌊\frac{deg(ρ)}{2}⌋.

3. Bezoutians

From now on we assume that the field E has characteristic different from 2. We study a quadratic form given by two polynomials f, g ∈ E[t] where g is squarefree and f is coprime to g. This quadratic form is obtained by a transfer and it is closely related to the Bezoutian of the polynomials f and g, introduced by Sylvester and Cayley. We shall develop some rules of computation, including a reciprocity law (Theorem 3.8).

We start by recalling some basic concepts and terminology from quadratic form theory. For a non-degenerate quadratic form φ over E, we denote by [φ] its Witt equivalence class, which we view as an element of the Witt ring W E.

For α ∈ W E we denote by diman(α), the dimension of the unique anisotropic quadratic form φ over E such that α = [φ], and call this the anisotropic dimension of α. In other terms, given a quadratic form φ over E, we have

dim(φ) = diman([φ]) + 2i_{W} (φ)

where dim(φ) and i_{W} (φ) are the dimension and the Witt index of φ. Unlike the dimension, the anisotropic dimension of a non-degenerate quadratic form depends only on its Witt equivalence class. Recall that the fundamental ideal IE consists of the classes [φ] of even-dimensional quadratic forms φ over E. We denote by I^{2}E the square of this ideal. For a ∈ E^{×} we denote by [a] the class in W E of the one-dimensional quadratic form ⟨a⟩ over E. Since quadratic forms over E are diagonalisable, W E is additively generated by the elements [a] with a ∈ E^{×}. Note that the unity in W E is the Witt equivalence class [1], which we simply denote by 1. The element 0 in W E is given by the trivial quadratic form. A quadratic form over E is said to be split if it is either hyperbolic or Witt equivalent to a one-dimensional quadratic form. Recall that the discriminant of a quadratic form φ over E is defined as the class of (−1)^{\frac{n(n−1)}{2}} det(M) in E^{×}/E^{×2} where n = dim(φ) and where M is the matrix of φ with respect to an arbitrary E-basis. The discriminant is an invariant of the Witt equivalence class of a form.

Let g ∈ E[t] be square-free. Let θ denote the class of t in E_{g} = E[t]/(g), n = deg(g), and let s_{g} : E_{g} ⟶ E be the E-linear form with s_{g}(θ^{i}) = 0 for i = 0, …, n − 2 and s_{g}(θ^{n−1}) = 1.

3.1. Proposition. Let f, g ∈ E[t] be coprime and θ = t+(g) in E_{g}. The quadratic map

q : E_{g} ⟶ E, x ↦ s_{g}(f(θ)x^{2})

is non-degenerate. The quadratic form (E_{g}, q) has discriminant N_{E_{g}/E}(f(θ))E^{×2}. The Witt index of this form is at least ⌊\frac{deg(g)−deg(f)}{2}⌋. In particular, if f is constant then (E_{g}, q) is split.

Proof. Set n = deg(g). Let B = (b_{ij})_{i,j} ∈ 𝕄_{n}(E) be given by

f(θ)θ^{j−1} = ∑_{i=1}^{n} b_{ij}θ^{i−1} .

For j, k = 1, …, n we obtain that

s_{g}(f(θ)θ^{j−1}θ^{k−1}) = ∑_{i=1}^{n} b_{ij}s_{g}(θ^{i−1}θ^{k−1}).

With C_{1} = (s_{g}(θ^{i−1}θ^{j−1}))_{i,j} ∈ 𝕄_{n}(E) and C_{f} = (s_{g}(f(θ)θ^{i−1}θ^{j−1}))_{i,j} ∈ 𝕄_{n}(E) we obtain that C_{f} = C_{1} · B and in particular

det(C_{f}) = det(C_{1}) · det(B) .

Note that N_{E_{g}/E}(f(θ)) = det(B) and

C_{1} = (\begin{matrix}0 & … & 0 & 1 \\ ⋮ & ... & ... & ∗ \\ 0 & ... & ... & ⋮ \\ 1 & ∗ & … & ∗ \end{matrix})

in view of the definition of s_{g}. Therefore

det(C_{f}) = (−1)^{⌊ \frac{n}{2} ⌋} · N_{E_{g}/E}(f(θ)) .

As C_{f} is the matrix of the quadratic form q with respect to the basis 1, θ, …, θ^{n−1}, this confirms that the quadratic form (E_{g}, q) is non-degenerate of discriminant N_{E_{g}/E}(f(θ))E^{×2} ∈ E^{×}/E^{×2}.

Furthermore, for k = ⌊\frac{deg(g)−deg(f)}{2}⌋ the vectors 1, …, θ^{k−1} span a totally isotropic subspace of (E_{g}, q), so the Witt index of (E_{g}, q) is at least k. □

We call the quadratic form in Proposition 3.1 the Bezoutian form of f modulo g. We are indebted to J.-P. Tignol for pointing out to us a different way of obtaining this quadratic form.

3.2. Proposition (Tignol). Let f, g ∈ E[t] be coprime such that g is monic and square-free and deg(f) ⩽ deg(g). Set θ = t + (g) in E_{g}. For n = deg(g) let A = (a_{ij})_{i,j} ∈ 𝕄_{n}(E) be the symmetric matrix given by

\frac{g(X)f(Y) − f(X)g(Y)}{X − Y} = ∑_{i,j=1}^{n} a_{ij}X^{i−1}Y^{j−1}.

Then the quadratic forms

E_{g} → E: x ↦ s_{g}(f(θ)x^{2}) and E^{n} → E: u ↦ u^{t}Au

are isometric.

Proof. We factorize

g(X) = (X − θ)h(X)

with h(X) ∈ E_{g}[X]. By definition of A we have

∑_{i,j=1}^{n} a_{ij}X^{i−1}θ^{j−1} = \frac{g(X)f(θ) − g(θ)f(X)}{X − θ} = h(X)f(θ).

We extend s_{g} to an E[X]-linear map E_{g}[X] → E[X]. We can write

h(X) = X^{n−1} + h_{1}(θ)X^{n−2} + ⋯ + h_{n−2}(θ)X + h_{n−1}(θ)

with polynomials h_{1}, …, h_{n−1} ∈ E[Y] of degree at most n − 1. Since g ∈ E[X] we obtain recursively for i = 1, …, n − 1 that h_{i} is monic of degree i. Hence

s_{g}(h(X)) = s_{g}(h_{n−1}(θ)) = 1 .

For any positive integer k we have

∑_{i,j=1}^{n} a_{ij}f(θ)^{−1}X^{i−1}θ^{j+k−2} = h(X)θ^{k−1}.

If 1 < k ⩽ n, then

(X^{k−1} − θ^{k−1})h(X) = g(X)(X^{k−2} + X^{k−3}θ + … + Xθ^{k−3} + θ^{k−2})

and s_{g} maps the right side to 0, showing that

∑_{i,j=1}^{n} a_{ij}s_{g}(f(θ)^{−1}θ^{j+k−2})X^{i−1} = s_{g} (h(X)θ^{k−1}) = s_{g} (h(X)X^{k−1}) = X^{k−1}.

The last equality also holds for k = 1. Hence, for any i, k ∈ {1, …, n}, we have

∑_{j=1}^{n} a_{ij}s_{g}(f(θ)^{−1}θ^{j+k−2}) = \begin{cases}1 & if i = k \\ 0 & if i ≠ k. \end{cases}

This shows that A is the inverse of (s_{g}(f(θ)^{−1}θ^{i+j−2}))_{i,j}, which is the representing matrix of q with respect to the E-basis (f(θ)^{−1}θ^{i})_{i=0}^{n−1} of E_{g}. Finally, since A is symmetric, it is congruent to A^{−1}. This proves the statement. □

More generally, for any two coprime polynomials f, g ∈ E[t], denoting by n the maximum of deg(f) and deg(g) the matrix A ∈ 𝕄_{n}(E) given by the formula in Proposition 3.2 defines a non-degenerate n-dimensional quadratic form over E, called the Hankel-Bezoutian of f and g. This quadratic form has interesting properties and seems to have been studied considerably in the literature (cf. [3]). The Bezoutian form of f modulo g coincides with the Hankel-Bezoutian of f and g in the situation of Proposition 3.2. In general, the two concepts behave differently.

Given two polynomials f, g ∈ E[t] with g square-free and prime to f, we write

𝔅 (\frac{f}{g} )

for the class in W E given by the Bezoutian form of f modulo g.

3.3. Proposition. For any a ∈ E^{×} and f, g, h ∈ E[t] with g square-free and coprime to fh, one has

𝔅 (\frac{afh^{2}}{g} ) = [a] · 𝔅 (\frac{f}{g} ) .

Proof. This follows directly from the properties of the transfer by which 𝔅 (\frac{∗}{g}) is defined. □

3.4. Proposition. Assume that the field E is infinite. Let f, g ∈ E[t] be coprime with g monic and square-free. Let θ = t+(g) in E_{g}. The following are equivalent:

(i) diman (𝔅 (\frac{f}{g} )) < deg(g). (ii) There exists f′ ∈ E[t] such that deg(f′) ⩽ deg(g) − 2 and f(θ)f′(θ) ∈ E_{g}^{×2}. (iii) There exists f′ ∈ E[t] square-free such that deg(f′) ⩽ deg(g) − 2,

deg(f′) ≡ deg(g) mod 2 and f(θ)f′(θ) ∈ E_{g}^{×2}.

Proof. Let n = deg(g). Let Ẽ be an algebraic closure of E and Ẽ_{g} = Ẽ[t]/(g). The equation

α = ∑_{i=1}^{n} c_{i}(α)θ^{n−i} for α ∈ Ẽ_{g}

determines Ẽ-linear forms c_{1}, …, c_{n} : Ẽ_{g} ⟶ Ẽ. For i = 1, …, n we obtain a quadratic form q_{i} : Ẽ_{g} ⟶ Ẽ, x ↦ c_{i}(f(θ)x^{2}). The quadratic space (E_{g}, q_{1}|_{E_{g}})

over E is non-degenerate and its Witt equivalence class is 𝔅 (\frac{f}{g} ). Hence, Condition (i) is equivalent to saying that (E_{g}, q_{1}|_{E_{g}}) is isotropic. In particular, (ii) implies (i). Trivially (iii) implies (ii). It remains to show that (i) implies (iii).

Suppose that there exists x ∈ E_{g}^{×} with q_{1}(x) = 0 ≠ q_{2}(x). Then α = f(θ)x^{2} ∈ E_{g}^{×} and c_{1}(α) = 0 ≠ c_{2}(α), whereby α = h(θ) for a polynomial h ∈ E[t] coprime to g with deg(h) = n − 2. We write h = f′ · ℓ^{2} with f′, ℓ ∈ E[t] such that f′ is square-free. Then f(θ)f′(θ) ∈ E_{g}^{×2} and deg(f′) = n − 2 − 2 deg(ℓ), whereby deg(f′) ⩽ deg(g) − 2 and deg(f′) ≡ deg(g) mod 2.

Hence, we are left with showing that the isotropy of (E_{g}, q_{1}|_{E_{g}}) implies the existence of x ∈ E_{g}^{×} with q_{1}(x) = 0 ≠ q_{2}(x).

Suppose first that n = 2. We choose any x ∈ E_{g} \ {0} with q_{1}(x) = 0. As E_{g} is reduced and f(θ) ∈ E_{g}^{×}, it follows that f(θ)x^{2} ∈ E^{×}, whereby x ∈ E_{g}^{×} and q_{2}(x) ≠ 0. Assume now that n ⩾ 3. Then q_{1} is irreducible. Consider the affine quadrics Q = {x ∈ Ẽ_{g} | q_{1}(x) = 0} and Q′ = {x ∈ Ẽ_{g} | q_{2}(x) = 0}. It follows that Q is irreducible as a topological space. We choose β ∈ Ẽ_{g} with c_{1}(β) = 0 ≠ c_{2}(β). As Ẽ_{g}^{×} = Ẽ_{g}^{×2}, there exists y ∈ Ẽ_{g} such that β = f(θ)y^{2}, and then q_{1}(y) = 0 ≠ q_{2}(y). Hence Q ⊈ Q′, whereby Q \ Q′ is a non-empty Zariski open subset of Q. The complement of Ẽ_{g}^{×} in Ẽ_{g} is a union of n affine hyperplanes, and therefore does not contain Q. Hence Ẽ_{g}^{×} ∩ Q is a non-empty Zariski open subset of Q. As Q is irreducible we conclude that (Q\Q′)∩(Ẽ_{g}^{×}∩Q) = Ẽ_{g}^{×}∩(Q\Q′) is also a non-empty Zariski open subset of Q. As (E_{g}, q_{1}) is isotropic, Q is rational over E. It follows that the set of E-rational points of Q is dense in Q. Hence Ẽ_{g}^{×}∩(Q\Q′) contains an E-rational point. In other words, there exists an element x ∈ E_{g}^{×} ∩ (Q \ Q′), and then q_{1}(x) = 0 ≠ q_{2}(x). □

In the sequel we give formulae for computing Bezoutians 𝔅 (\frac{f}{g} ) where both f, g ∈ E[t] are monic and square-free. Together with Proposition 3.3 this yields means of computation for more general situations.

3.5. Proposition. For f, g_{1}, g_{2} ∈ E[t] pairwise coprime and with g_{1} and g_{2} monic and square-free, we have that

𝔅 (\frac{f}{g_{1}g_{2}} ) = 𝔅 (\frac{fg_{2}}{g_{1}} ) + 𝔅 (\frac{fg_{1}}{g_{2}} ) .

Proof. Set g = g_{1}g_{2}. Let θ, θ_{1}, θ_{2} be the images of t in E_{g}, E_{g_{1}} and E_{g_{2}} respectively. Define the map Ψ : E_{g_{1}} ⊕ E_{g_{2}} ↦ E_{g} by

Ψ(h_{1}(θ_{1}) + h_{2}(θ_{2})) = (h_{1}g_{2} + h_{2}g_{1})(θ).

Then Ψ is a E-vector space isomorphism. For i = 1, 2 one has

s_{g_{i}} = (s_{g} ∘ Ψ)|_{E_{i}} .

We consider the quadratic maps

q: E_{g} → E: x ↦ s_{g}(f(θ)x^{2}) and
q_{i}: E_{g_{i}} → E: x_{i} ↦ s_{g_{i}}((fg_{3−i})(θ_{i})x_{i}^{2}) for i = 1, 2.

For x_{1} = h_{1}(θ_{1}) and x_{2} = h_{2}(θ_{2}) we have Ψ(x_{1}) = (h_{1}g_{2})(θ), Ψ(x_{2}) = (h_{2}g_{1})(θ) and Ψ(x_{1})Ψ(x_{2}) = 0, and therefore

q(Ψ(x_{1} + x_{2})) = s_{g}(f(θ)(Ψ(x_{1}) + Ψ(x_{2}))^{2})
= s_{g}(f(θ)Ψ(x_{1})^{2}) + s_{g}(f(θ)Ψ(x_{2})^{2}) + 2s_{g}(f(θ)Ψ(x_{1})Ψ(x_{2}))
= s_{g}((fh_{1}^{2}g_{2}^{2})(θ)) + s_{g}((fh_{2}^{2}g_{1}^{2})(θ))
= (s_{g} ∘ Ψ)((fg_{2}h_{1}^{2})(θ_{1})) + (s_{g} ∘ Ψ)((fg_{1}h_{2}^{2})(θ_{2}))
= s_{g_{1}}((fg_{2})(θ_{1}) · x_{1}^{2}) + s_{g_{2}}((fg_{1})(θ_{2}) · x_{2}^{2})
= q_{1}(x_{1}) + q_{2}(x_{2}).

Hence Ψ is an isometry between (E_{g_{1}}, q_{1}) ⊥ (E_{g_{2}}, q_{2}) and (E_{g}, q). □

3.6. Corollary. Let n ∈ ℕ and a_{1}, …, a_{n} ∈ E^{×} pairwise distinct and f ∈ E[t] such that f(a_{i}) ≠ 0 for i = 1, …, n. Then

𝔅 (\frac{f}{∏_{i=1}^{n}(t − a_{i})}) = ∑_{i=1}^{n} [f(a_{i}) ∏_{j≠i}(a_{i} − a_{j})].

Proof. For g = ∏_{i=1}^{n}(t − a_{i}) we obtain by an iterative application of Proposition 3.5 that

𝔅 (\frac{f}{g} ) = ∑_{i=1}^{n} 𝔅 (\frac{f ∏_{j≠i}(t − a_{j})}{t − a_{i}}) = ∑_{i=1}^{n} [f(a_{i}) ∏_{j≠i}(a_{i} − a_{j})].

□

3.7. Corollary. Let f_{1}, f_{2}, g_{1}, g_{2} ∈ E[t] with g_{1} and g_{2} monic, square-free, relatively coprime and such that f_{i} is coprime to g_{i} for i = 1, 2. Then

𝔅 (\frac{f_{1}}{g_{1}} ) + 𝔅 (\frac{f_{2}}{g_{2}} ) = 𝔅 (\frac{f_{1}g_{2} + f_{2}g_{1}}{g_{1}g_{2}} ) .

Proof. Let h = f_{1}g_{2} + f_{2}g_{1}. For i = 1, 2 we have that hg_{3−i} ≡ f_{i} g_{3−i}^{2} mod g_{i}. Using Proposition 3.5 and Proposition 3.3 we thus obtain that

𝔅 (\frac{h}{g_{1}g_{2}} ) = 𝔅 (\frac{hg_{2}}{g_{1}} ) + 𝔅 (\frac{hg_{1}}{g_{2}} ) = 𝔅 (\frac{f_{1}}{g_{1}} ) + 𝔅 (\frac{f_{2}}{g_{2}} ) .

□

The Bezoutians studied here satisfy a reciprocity law, which will turn out to be very useful in the sequel of the article.

3.8. Theorem. Let f, g ∈ E[t] be monic, square-free and relatively coprime. Then

𝔅 (\frac{f}{g} ) + 𝔅 (\frac{g}{f} ) = \{ \begin{matrix}0 & if deg(f) ≡ deg(g) mod 2, \\ 1 & if deg(f) ≢ deg(g) mod 2. \end{matrix}

Proof. It follows using Corollary 3.7 and Proposition 3.3 that

𝔅 (\frac{f}{g} ) + 𝔅 (\frac{g}{f} ) = 𝔅 (\frac{f^{2} + g^{2}}{fg} ) = 𝔅 (\frac{(f + g)^{2}}{fg} ) = 𝔅 (\frac{1}{fg} ) .

By Proposition 3.1 this element of W E is given by the split quadratic form of dimension deg(fg) and of trivial discriminant. This yields the statement. □

Given two elements α, β ∈ W E, we write α ∼ β if β = cα for some c ∈ E^{×}, that is, if α and β are given by quadratic forms over E that are similar.

3.9. Corollary. Let f, g ∈ E[t] be square-free, relatively coprime and such that deg(f) ≡ deg(g) mod 2. Then

𝔅 (\frac{f}{g} ) ∼ 𝔅 (\frac{g}{f} ) .

Proof. Let a, b ∈ E^{×} and f′, g′ ∈ E[t] monic such that f = af′ and g = bg′. Using Theorem 3.8 we obtain that

𝔅 (\frac{f}{g} ) = [a] · 𝔅 (\frac{f′}{g′} ) = [−a] · 𝔅 (\frac{g′}{f′} ) = [−ab] · 𝔅 (\frac{g}{f} ) .

□

4. Bezoutians related to a symbol

We want to compute the Bezoutian of two coprime polynomials that come from a particular representation of a symbol.

4.1. Proposition. Let σ be a symbol in k_{2}E(t). Then there exist f, g ∈ E[t] square-free, coprime and with g of even degree such that

σ = {f, g}

and ∂_{p}(σ) = 0 for every prime factor p of f.

Proof. We choose q, r, s ∈ E[t] square-free and pairwise coprime with σ = {qs, rs}. Then

σ = {qs + rs, −qr} = {qs + rs − qr, (q + r)qrs} .

We thus have σ = {f′, g′} for two coprime polynomials f′, g′ ∈ E[t] such that g′ is a multiple of qrs. We may assume that f′, g′ and f′(1 − g′) are all different from 1, because otherwise σ = 0 and the statement holds trivially. Using now that 0 = {f′, 1 − f′} = {1 − g′, g′} = {f′(1 − g′), 1 − f′(1 − g′)}, we rewrite

σ = {f′, g′} = {f′, g′(1 − f′)} = {f′(1 − g′), (1 − f′(1 − g′))g′} .

At least one of the polynomials g′, g′(1−f′) and (1−f′(1−g′))g′ has even degree. Hence, in any case we obtain a representation

σ = {f′′, g′′}

with f′′, g′′ ∈ E[t] coprime and where g′′ is of even degree and a multiple of qrs. We write f′′ = fh^{2} and g′′ = gh^{′2} with f, g ∈ E[t] square-free and h, h′ ∈ E[t]. Then f and g are coprime and square-free, g has even degree and

σ = {f, g} .

Furthermore, f is coprime to g′′ and thus to qrs. Since σ = {qs, rs} we conclude that ∂_{p}(σ) = 0 for every p ∈ P not dividing qrs, so in particular for every prime factor of f. □

4.2. Lemma. Let f, g, h ∈ E[t] be monic, square-free and pairwise coprime, and let a, b ∈ E^{×} be such that ∂_{p}({af, bgh}) = 0 for every p ∈ P not dividing g. Then

𝔅 (\frac{af}{g} ) = \{ \begin{matrix}0 & if & (deg(f), deg(g), deg(h)) ≡ (0, 0, 0) mod 2, \\ [b] − [ab] & if & (deg(f), deg(g), deg(h)) ≡ (0, 0, 1) mod 2, \\ [a] & if & (deg(f), deg(g), deg(h)) ≡ (0, 1, 0) mod 2, \\ [a] + [b] − [ab] & if & (deg(f), deg(g), deg(h)) ≡ (0, 1, 1) mod 2, \\ [a] − [ab] & if & (deg(f), deg(g), deg(h)) ≡ (1, 0, 0) mod 2, \\ [a] + [b] & if & (deg(f), deg(g), deg(h)) ≡ (1, 0, 1) mod 2, \\ −[ab] & if & (deg(f), deg(g), deg(h)) ≡ (1, 1, 0) mod 2, \\ [b] & if & (deg(f), deg(g), deg(h)) ≡ (1, 1, 1) mod 2, \end{matrix}

Furthermore, if ∂_{∞}({af, bgh}) = 0, then

𝔅 (\frac{af}{g} ) = \{ \begin{matrix}0 & if deg(g) is even, \\ [a] + [b] − [ab] & otherwise. \end{matrix}

Proof. We have

𝔅 (\frac{af}{g} ) = [a] · 𝔅 (\frac{f}{g} ) = −[a] · 𝔅 (\frac{g}{f} ) + \{ \begin{matrix}0 & if deg(fg) is even, \\ [a] & otherwise. \end{matrix}

Set σ = \{af, bgh\}. For every p ∈ P dividing f we have \{\overline{bgh}\} = ∂_{p}(σ) = 0 in k_{1}E_{p}. Hence, bgh is a square modulo f, whereby

𝔅 (\frac{g}{f} ) = [b] · 𝔅 (\frac{h}{f}) = −[b] · 𝔅 (\frac{f}{h}) + \{ \begin{matrix}0 & if deg(fh) is even, \\ [b] & otherwise. \end{matrix}

For every p ∈ P dividing h we have {af} = ∂_{p}(σ) = 0. Hence af is a square modulo h, whereby

𝔅 (\frac{f}{h}) = 𝔅 (\frac{a}{h}) = \{ \begin{matrix}0 & if deg(h) is even, \\ [a] & otherwise. \end{matrix}

This together shows the first part of the statement.

Note further that

∂_{∞}(σ) = \{ \begin{matrix}0 & if (deg(f), deg(gh)) ≡ (0, 0) mod 2, \\ \{a\} & if (deg(f), deg(gh)) ≡ (0, 1) mod 2, \\ \{b\} & if (deg(f), deg(gh)) ≡ (1, 0) mod 2, \\ \{−ab\} & if (deg(f), deg(gh)) ≡ (1, 1) mod 2. \end{matrix}

Suppose now that ∂_{∞}(σ) = 0. In each case where f or gh has odd degree, we conclude that at least one of a, b and −ab is a square, that 𝔅 (\frac{af}{g} ) is split of trivial discriminant and that [1] − [a] − [b] + [ab] = [⟨1, −a, −b, ab⟩] = 0. This yields the second part of the statement. □

4.3. Example. Let E = ℚ_{p} for a prime number p with p ≡ 5 mod 8. Hence −1 ∈ ℚ_{p}^{×2} and 2 ∉ ℚ_{p}^{×2}. For a = 2 and b = p, the polynomials f = t(t − 2), g = (t−1)(t−p)(t+2p−2) and h = t−p+2 satisfy the conditions in Lemma 4.2 with (deg(f), deg(g), deg(h)) ≡ (0, 1, 1) mod 2. In this case 𝔅 (\frac{af}{g} ) is given by the anisotropic quadratic form ⟨2, 2p, p⟩.

4.4. Proposition. Let f, g, h ∈ E[t] be square-free and pairwise coprime with deg(g) even and such that ∂_{p}({f, gh}) = 0 for all p ∈ P not dividing g as well as for p = ∞. Then

𝔅 (\frac{f}{g} ) = 0 .

Proof. This follows from Lemma 4.2. □

It follows from Proposition 4.1 that any symbol σ ∈ k_{2}E(t) with ∂_{∞}(σ) = 0 has representations σ = {f, g} = {f, gh} with f, g ∈ E[t] and h = 1 such that the conditions in Proposition 4.4 are satisfied.

5. Splitting sequences

Given ramification sequence ρ, we shall now construct a sequence of polynomials that yields a representation of ρ as the ramification of a sum of symbols. This leads to an improved bound on the length of such a representation of ρ.

Given g ∈ E[t] monic and square-free and f ∈ E[t] coprime to g, we denote by ℜ (\frac{f}{g} ) the element ρ ∈ ℜ_{2}(E) defined by ρ_{p} = \{\overline{f}\} for p ∈ P dividing g and by ρ_{p} = 0 for all other p ∈ P.

For a finite subset S ⊆ P we set

P_{S} = ∏_{p∈S} p ∈ E[t]

and observe that this polynomial is monic and square-free.

Consider a ramification sequence ρ ∈ ℜ_{2}(E). We fix a finite subset S ⊆ P such that Supp(ρ) ⊆ S ∪ {∞}. We set deg(S) = _{∑p∈S} deg(p) and g = P_{S}. Then ℜ (\frac{f}{g} ) = ρ for a polynomial f ∈ E[t] coprime to g which is determined by S up to a square modulo g. We set

𝔅_{S}(ρ) = 𝔅 (\frac{f}{g} ) .

The aim of this section is to relate ρ to 𝔅_{S}(ρ).

5.1. Proposition. We have the following:

(a) The discriminant of 𝔅_{S}(ρ) is the square class corresponding to ρ_{∞} ∈ k_{1}E.

(b) 𝔅_{S}(ρ) ∈ IE if and only if deg(S) is even.

(c) 𝔅_{S}(ρ) ∈ I^{2}E if and only if deg(S) is even and ρ_{∞} = 0.

Proof. Part (b) is obvious and (c) follows from (a) and (b) by using [5, Chap. II, Corollary 2.2]. It remains to show (a). Since ρ ∈ ℜ_{2}(E) = ker(N), we obtain in k_{1}E that

0 = N(ρ) = ∑_{p∈Supp(ρ)} N_{E_{p}/E}(ρ_{p}) = ρ_{∞} + ∑_{p∈S} N_{E_{p}/E}(ρ_{p}) .

With g = P_{S} and f ∈ E[t] coprime to g such that ρ = ℜ (\frac{f}{g} ), it follows that

ρ_{∞} = ∑_{p∈S} N_{E_{p}/E}(f + (p)) = N_{E_{g}/E}(f + (g)) .

By Proposition 3.1 the last term is given by the discriminant of 𝔅 (\frac{f}{g} ) = 𝔅_{S}(ρ). □

5.2. Proposition. Let r ∈ ℕ, f_{0}, …, f_{r} ∈ E[t] square-free with f_{r} ∈ E^{×} and such that, with f_{r+1} = 1, we have that f_{i−1} and f_{i} are relatively coprime and f_{i−1}f_{i+1} is a square modulo f_{i} for i = 1, …, r. Then

ℜ (\frac{f_{1}}{f_{0}} ) = ∂ (∑_{k=1}^{⌈ \frac{r}{2} ⌉} \{f_{2k−2}f_{2k}, f_{2k−1}\}) .

Furthermore, for s = |{i ∈ {1, …, r} | deg(f_{i}) ≢ deg(f_{i−1}) mod 2}| we have

diman (𝔅 (\frac{f_{1}}{f_{0}} )) ⩽ s ⩽ r and diman 𝔅 (\frac{f_{1}}{f_{0}} ) ≡ s mod 2 .

Proof. For i = 1, …, r we have that ℜ (\frac{f_{i−1}f_{i+1}}{f_{i}} ) = 0 because f_{i−1}f_{i+1} is a square

modulo f_{i}. In particular ℜ (\frac{f_{r−1}}{f_{r}} ) = ℜ (\frac{f_{r+1}}{f_{r}} ) = 0, as f_{r+1} = 1.

With n = ⌈\frac{r}{2}⌉ we obtain from the conditions that

∂ (∑_{k=1}^{n} {f_{2k−2}f_{2k}, f_{2k−1}}) = ∂ (∑_{i=1}^{r} {f_{i−1}, f_{i}})

= ∑_{i=1}^{r} (ℜ (\frac{f_{i−1}}{f_{i}} ) + ℜ (\frac{f_{i}}{f_{i−1}} ))

= ∑_{i=1}^{r} ℜ (\frac{f_{i−1}}{f_{i}} ) + ∑_{i=0}^{r−1} ℜ (\frac{f_{i+1}}{f_{i}} )

= ℜ (\frac{f_{1}}{f_{0}} ) + (∑_{i=1}^{r−1} ℜ (\frac{f_{i−1}f_{i+1}}{f_{i}} )) + ℜ (\frac{f_{r−1}}{f_{r}} )

= ℜ (\frac{f_{1}}{f_{0}} ) .

We prove the second part of the statement by induction on r. If r = 0 then f_{1} = 1 and diman(𝔅 (\frac{f_{1}}{f_{0}} )) = deg(f_{0}) = s = 0 and the claim holds trivially. Suppose now that r > 0. For s′ = |{i ∈ {2, …, r} | deg(f_{i}) ≢ deg(f_{i−1}) mod 2}| we obtain by the induction hypothesis that

diman 𝔅 (\frac{f_{2}}{f_{1}} ) ⩽ s′ ⩽ r − 1 and diman 𝔅 (\frac{f_{2}}{f_{1}} ) ≡ s′ mod 2 .

If deg(f_{0}) ≡ deg(f_{1}) mod 2, then s = s′ and

𝔅 (\frac{f_{1}}{f_{0}} ) ∼ 𝔅 (\frac{f_{0}}{f_{1}} ) = 𝔅 (\frac{f_{2}}{f_{1}} ) .

If deg(f_{0}) ≢ deg(f_{1}) mod 2, then s = s′ + 1 and

𝔅 (\frac{f_{1}}{f_{0}} ) ∼ 𝔅 (\frac{f_{0}}{f_{1}} ) + [c] = 𝔅 (\frac{f_{2}}{f_{1}} ) + [c] ,

for some c ∈ E^{×}. In either case we conclude that

diman 𝔅 (\frac{f_{1}}{f_{0}} ) ⩽ s ⩽ r and diman 𝔅 (\frac{f_{1}}{f_{0}} ) ≡ s mod 2 . □

For r ∈ ℕ and f_{0}, …, f_{r} such as in Proposition 5.2 and with ℜ (\frac{f_{1}}{f_{0}} ) = ρ, we call (f_{0}, …, f_{r}) a splitting sequence of ρ, and refer to the number r as its length. We say that the splitting sequence (f_{0}, …, f_{r}) is strictly decreasing if we have deg(f_{i}) < deg(f_{i−1}) for i = 1, …, r.

5.3. Corollary. Let r be the length of a splitting sequence of ρ. Then ρ is represented by a sum of n symbols for n = ⌈\frac{r}{2}⌉.

Proof. This is immediate from Proposition 5.2. □

One may ask whether any representation of ρ by a given number of symbols can be obtained from a splitting sequence via Proposition 5.2.

Having thus shown that a splitting sequence gives rise to a representation of ρ by a sum of symbols, we turn to the problem of constructing splitting sequences of small length. Recall that we are assuming throughout that Supp(ρ) ⊆ S ∪ {∞}.

5.4. Proposition. There exists a strictly decreasing splitting sequence of ρ starting with P_{S}. The length of any such splitting sequence is at most deg(S).

Proof. We set f_{0} = P_{S}. If f_{0} = 1 then (f_{0}) is the desired sequence. Otherwise deg(f_{0}) > 0 and we choose f_{1} ∈ E[t] square-free and coprime to f_{0} such that deg(f_{1}) < deg(f_{0}) and ℜ (\frac{f_{1}}{f_{0}} ) = ρ. For i ⩾ 1 with deg(f_{i−1}) > 0 we choose recursively f′_{i} ∈ E[t] as the unique polynomial with deg(f′_{i}) < deg(f_{i−1}) and f′_{i} ≡ f_{i−2} mod f_{i−1} and then let f_{i} denote the square-free part of f′_{i}, and we stop the process and set r = i − 1 as soon as deg(f_{i−1}) = 0. In this way we obtain a strictly decreasing splitting sequence of ρ starting with P_{S}.

Suppose now that r ∈ ℕ is the length of a strictly decreasing splitting sequence of ρ starting with P_{S}, say (f_{0}, …, f_{r}). As deg(f_{i−1}) − deg(f_{i}) ⩾ 1 for i = 1, …, r and further deg(f_{r}) = 0, we conclude that

deg(S) = deg(f_{0}) − deg(f_{r}) = ∑_{i=1}^{r} (deg(f_{i−1}) − deg(f_{i})) ⩾ r .
□

5.5. Corollary. If diman 𝔅_{S}(ρ) = deg(S), then the length of any strictly decreasing splitting sequence of ρ starting with P_{S} is equal to deg(S).

Proof. Let g = P_{S} and choose f ∈ E[t] coprime to g with ℜ (\frac{f}{g} ) = ρ. Let r ∈ ℕ be the length of a strictly decreasing splitting sequence for ρ starting with g. As 𝔅_{S}(ρ) = 𝔅 (\frac{f}{g} ) we obtain that diman 𝔅_{S}(ρ) ⩽ r ⩽ deg(S) by Proposition 5.2 and Proposition 5.4. Hence, if diman 𝔅_{S}(ρ) = deg(S) then r = deg(S). □

We give an example where the hypothesis in Corollary 5.5 is satisfied for an arbitrary given degree.

5.6. Example. Let E_{0} be a field, d ⩾ 3 and a_{1}, …, a_{d} ∈ E_{0}^{×} pairwise different. Let b_{1}, …, b_{d−1} be indeterminates over E_{0} and set E = E_{0}((b_{1})) … ((b_{d−1})) and b_{d} = b_{1} ⋯ b_{d−1}. Consider

ξ = ∑_{i=1}^{d}{(1 − a_{i})t + a_{i}, b_{i}}

in k_{2}E(t) and ρ = ∂(ξ). Note that Supp(ρ) = \{t − \frac{a_{i}}{a_{i}−1} | i = 1, …, d\} and thus deg(ρ) = d. Using Corollary 3.6 we obtain for S = Supp(ρ) that

𝔅_{S}(ρ) = ∑_{i=1}^{d}[b_{i}c_{i}]

for certain c_{1}, …, c_{d} ∈ E_{0}^{×}. It follows that diman 𝔅_{S}(ρ) = d = deg(S).

5.7. Lemma. Assume that E is infinite. Let g ∈ E[t] be square-free and f ∈ E[t] coprime to g. There exist m ∈ ℕ and square-free polynomials f_{1}, …, f_{m} ∈ E[t] such that, with f_{0} = g and f_{−1} = f the following are satisfied for i = 1, …, m:

• deg(f_{i}) ⩽ deg(f_{i−1}) − 2 and deg(f_{i}) ≡ deg(g) mod 2;

• f_{i−1} and f_{i} are relatively coprime;
• f_{i}f_{i−2} is a square modulo f_{i−1};
• deg(f_{m}) = diman 𝔅 (\frac{f}{g} ).

Furthermore, under these conditions

m ⩽ \frac{1}{2} (deg(g) − diman 𝔅 (\frac{f}{g} )) and 𝔅 (\frac{f_{i−1}}{f_{i}} ) ∼ 𝔅 (\frac{f}{g} ) for i = 1, …, m.

Proof. Assume r ∈ ℕ and polynomials f_{0}, …, f_{r} with f_{0} = g are given satisfying all of the given conditions not involving diman 𝔅 (\frac{f}{g} ). Then by Corollary 3.9 we have

𝔅 (\frac{f_{i−1}}{f_{i}} ) ∼ 𝔅 (\frac{f_{i}}{f_{i−1}} ) = 𝔅 (\frac{f_{i−2}}{f_{i−1}} ) for i = 1, …, r .

Thus 𝔅 (\frac{f_{i−1}}{f_{i}} ) ∼ 𝔅 (\frac{f}{g} ) and diman 𝔅 (\frac{f_{i−1}}{f_{i}} ) = diman 𝔅 (\frac{f}{g} ) for i = 1, …, r. Furthermore, as deg(f_{i−1}) − deg(f_{i}) ⩾ 2 for i = 1, …, r and f_{0} = g we obtain that

deg(g) − deg(f_{r}) = (∑_{i=1}^{r}(deg(f_{i−1}) − deg(f_{i}))) ⩾ 2r.

As deg(f_{r}) ⩾ diman 𝔅 (\frac{f_{r−1}}{f_{r}} ) = diman 𝔅 (\frac{f}{g} ), we conclude that

r ⩽ \frac{1}{2} (deg(g) − diman 𝔅 (\frac{f}{g} )) .

This shows the last part of the statement.

We now construct recursively a sequence of square-free polynomials as claimed. We set f_{−1} = f and f_{0} = g. For i ⩾ 1, as long as deg(f_{i−1}) > diman 𝔅 (\frac{f_{i−2}}{f_{i−1}} ) holds, we use Proposition 3.4 to obtain a square-free polynomial f_{i} ∈ E[t] coprime to f_{i−1} such that deg(f_{i}) ⩽ deg(f_{i−1}) − 2, deg(f_{i}) ≡ deg(f_{i−1}) mod 2 and f_{i}f_{i−2} is a square modulo f_{i−1}. As soon as deg(f_{i−1}) = diman 𝔅 (\frac{f_{i−2}}{f_{i−1}} ) holds, we stop and set m = i − 1. It follows from what we showed in the first part of the proof that the recursion terminates after finitely many steps. □

5.8. Proposition. Assume that E is infinite. Set d = diman 𝔅_{S}(ρ). There exist m ∈ ℕ and a strictly decreasing splitting sequence (f_{0}, …, f_{d+m}) of ρ starting with P_{S} such that deg(f_{i}) ≡ deg(f_{i−1}) mod 2 for i = 1, …, m and deg(f_{m+d−i}) = i for i = 0, …, d.

Proof. Let f_{0} = P_{S} and f_{−1} ∈ E[t] coprime to f_{0} with ℜ (\frac{f_{−1}}{f_{0}} ) = ρ. We choose m ∈ ℕ and f_{1}, …, f_{m} ∈ E[t] as in Lemma 5.7. Then deg(f_{m}) = d = diman 𝔅_{S}(ρ) and 𝔅 (\frac{f_{m−1}}{f_{m}} ) ∼ 𝔅 (\frac{f_{−1}}{f_{0}} ) = 𝔅_{S}(ρ). Set ρ′ = ℜ (\frac{f_{m−1}}{f_{m}} ). By Proposition 5.4 and Corollary 5.5 there exists a strictly decreasing splitting sequence of ρ′ of length d starting with f_{m}, say (f_{m}, …, f_{d+m}). Then (f_{0}, …, f_{d+m}) has the required properties. □

5.9. Theorem. Any splitting sequence ρ with Supp(ρ) ⊆ S ∪ {∞} is represented by a sum of n symbols where n = ⌈\frac{deg(S)+diman 𝔅_{S}(ρ)}{4}⌉ .

Proof. Set d = diman 𝔅_{S}(ρ). In view of Example 2.4 we may assume that E is infinite. Let m ∈ ℕ and let (f_{0}, …, f_{d+m}) be a strictly decreasing splitting sequence of ρ starting with P_{S} and with the properties formulated in Proposition 5.8. We have m ⩽ \frac{1}{2}(deg(S) − d), whence d + m ⩽ \frac{1}{2}(deg(S) + d). The statement thus follows from Corollary 5.3. □

5.10. Remark. By Theorem 2.5 the ramification sequence ρ is represented by a sum of n′ symbols for n′ = ⌊\frac{deg(ρ)}{2}⌋ symbols. Let us compare n′ with the value of n in Theorem 5.9 that we obtain for S = Supp(ρ) \ {∞}. Obviously

deg(S) = \begin{cases}deg(ρ) & if ρ_{∞} = 0, \\ deg(ρ) − 1 & if ρ_{∞} ≠ 0. \end{cases}

and diman 𝔅_{S}(ρ) ⩽ deg(S). We conclude that n ⩽ n′ except in the case where deg(S) is odd, diman 𝔅_{S}(ρ) = deg(S) and ρ_{∞} = 0.

Let us look more closely at this case. With d = deg(S) = diman(𝔅_{S}(ρ)) we obtain that n = \frac{d+1}{2} = n′ + 1. The strictly decreasing splitting sequence obtained from Lemma 5.7 which is used in the proof of Theorem 5.9 has length d. Denoting this sequence (f_{0}, …, f_{d}), we have that deg(f_{i}) = d − i for i = 1, …, d. Applying Proposition 5.2 we obtain that ρ = ∂(∑_{k=1}^{n} σ_{k}) where σ_{k} = {f_{2k−2}f_{2k}, f_{2k−1}} for k = 1, …, n and f_{d+1} = 1. As deg(f_{i}) ≡ i + 1 mod 2 for i = 0, …, d, it is easy to see that ∂_{∞}(σ_{k}) = 0 for k = 1, …, n − 1 and ∂_{∞}(σ_{n}) = ∂_{∞}({f_{d−1}, f_{d}}) = {f_{d}} in k_{1}E. Hence, {f_{d}} = ∂_{∞}(∑_{k=1}^{n} σ_{k}) = ρ_{∞} = 0, whereby f_{d} ∈ E^{×2}. This implies that σ_{n} = {f_{d−1}, f_{d}} = 0 in k_{2}E(t). We conclude that ρ = ∂(∑_{k=1}^{n−1} σ_{k}).

Hence, in each case our method applied as in the proof of Theorem 5.9 yields a representation of ρ by at most n′ symbols.

Note finally that if deg(ρ) is odd and diman 𝔅_{S}(ρ) = deg(ρ) − 4, then n′ = n. In particular, if deg(ρ) = 5 and ρ_{∞} = 0, then n ⩾ n′ = 2, and we shall see in Example 6.3 that even in the case where 𝔅_{S}(ρ) is split a single symbol may not be sufficient to represent ρ.

In relation to Theorem 5.9 one may ask the following:

5.11. Question. Let n ∈ ℕ be such that ρ is represented by a sum of n symbols. Set S = Supp(ρ) \ {∞}. Is there an upper bound on diman 𝔅_{S}(ρ) in terms of n?

The following result answers Question 5.11 partially in the case where n = 1.

5.12. Theorem. Set S = Supp(ρ) \ {∞} and assume that deg(ρ) is even. If ρ is represented by a symbol, then 𝔅_{S}(ρ) is split.

Proof. By Proposition 4.1 there exist square-free coprime polynomials f, g′ ∈ E[t] such that σ = {f, g′} and ∂_{p}(σ) = 0 for all p dividing f. It follows that g′ = gh with g = P_{S} and h ∈ E[t]. Then f, g, h ∈ E[t] are square-free and coprime and ∂_{p}(σ) = 0 holds for all p not dividing g. Then ∂(σ) = ℜ (\frac{f}{g} ) and 𝔅_{S}(ρ) = 𝔅 (\frac{f}{g} ). As deg(ρ) is even, the discriminant of 𝔅_{S}(ρ) is trivial if and only if deg(g) is even. By Lemma 4.2 we have (deg(f), deg(g), deg(h)) ≢ (0, 1, 1) mod 2 and if deg(g) is odd, then 𝔅 (\frac{f}{g} ) is split. If deg(g) is even, whereby the discriminant of 𝔅 (\frac{f}{g} ) is trivial, then Lemma 4.2 yields that 𝔅 (\frac{f}{g} ) = 0. □

6. Ramification sequences of small degree

We conclude our study with a statement on ramification sequences of degree four followed by a series of examples of ramification sequences of small degree.

6.1. Theorem. Assume that deg(ρ) = 4 and set S = Supp(ρ) \ {∞}. Then ρ is represented by a symbol if and only if 𝔅_{S}(ρ) is split.

Proof. If E is infinite and diman 𝔅_{S}(ρ) ⩽ 1 then we conclude by Theorem 5.9 that ρ = ∂(σ) for a symbol σ ∈ k_{2}E(t). If E is finite then the same conclusion holds trivially by Example 2.4. The converse follows from Theorem 5.12. □

6.2. Remark. Assume that deg(ρ) = 4 and let S = Supp(ρ) \ {∞}. If ρ_{∞} ≠ 0, then with d ∈ E^{×} such that ρ_{∞} = {d} in k_{1}E, it follows that [1]−[d]·𝔅_{S}(ρ) = [π] for a 2-fold Pfister form π over E. Then π is hyperbolic if and only if 𝔅_{S}(ρ) is split, and by Theorem 6.1 this is if and only if ρ is represented by a symbol. In the special case where S consists of three rational points, this characterisation was given in [8, Prop. 1.4] with an explicit construction of the form π, which can actually be retrieved from Corollary 3.6.

In the case where ρ_{∞} = 0, the criterion in Theorem 6.1 can be reformulated in similar terms. In this case there is a unique 2-fold Pfister form π over E such that 𝔅_{S}(ρ) ∼ [π], and ρ is represented by a symbol if and only if π is hyperbolic.

Let ρ ∈ ℜ_{2}(E). The following example shows that Theorem 6.1 does not extend to the case where deg(ρ) = 5.

6.3. Example. Set ξ = {(t − 1)(t + 1)(t − 2)(t − 3), −1} + {t(t − 1), 7} in k_{2}ℚ_{7}(t). Then S = Supp(∂(ξ)) consists of the polynomials t − m with m = −1, 0, 1, 2, 3. By [4, Theorem 3.6] we have ∂(ξ) ≠ ∂(σ) for any symbol σ in k_{2}ℚ_{7}(t). Using Corollary 3.6 we obtain that 𝔅_{S}(∂(ξ)) = [1], whence 𝔅_{S}(∂(ξ)) is split.

We are going to look at some examples where deg(ρ) = 6 and ρ_{∞} = 0 and where ρ is not represented by a symbol. In the first two examples we have that 𝔅_{S}(ρ) = 0 for S = Supp(ρ). In particular, the converse of the implication in Theorem 5.12 does not hold in general.

6.4. Example. We consider the field E = ℚ_{p} for a prime number p ⩾ 7. We find α, a, b ∈ ℤ \ pℤ such that a^{2} + 1 ∈ ℚ_{p}^{×2} and α, b^{2} + 1 ∉ ℚ_{p}^{×2}. Set

ξ = {(t^{2} − a^{2})(t^{2} − b^{2}), α} + {t^{2} + 1, p} ∈ k_{2}ℚ_{p}(t) .

By [4, Theorem 3.6] we have that ∂(ξ) ≠ ∂(σ) for all symbols σ ∈ k_{2}ℚ_{p}(t). We set ρ = ∂(ξ). Set S = Supp(ρ) and note that S consists of the prime factors of (t^{2} − a^{2})(t^{2} − b^{2})(t^{2} + 1). Hence deg(S) = 6. Computation yields that

𝔅_{S}(ρ) = [α] · 𝔅 (\frac{t^{2} + 1}{(t^{2} − a^{2})(t^{2} − b^{2})}) + [p] · 𝔅 (\frac{(t^{2} − a^{2})(t^{2} − b^{2})}{t^{2} + 1})
= [⟨−α, p⟩] · 𝔅 (\frac{(t^{2} − a^{2})(t^{2} − b^{2})}{t^{2} + 1})
= [⟨−α, p⟩] · 𝔅 (\frac{(1 + a^{2})(1 + b^{2})}{t^{2} + 1}) = 0

6.5. Example. Consider E = ℚ_{p} for a prime number p ≡ ±3 mod 8. Set

ξ = {t(t − 1)(t − 2)(t − 3), 2} + {t^{2} + 1, p} ∈ k_{2}ℚ_{p}(t).

By [4, Theorem 3.6] we have that ∂(ξ) ≠ ∂(σ) for any symbol σ ∈ k_{2}ℚ_{p}(t). Note that S = Supp(∂(ξ)) is the set of prime factors of g = t(t − 1)(t − 2)(t − 3)(t^{2} + 1) over ℚ_{p}. Thus deg(S) = 6. Using that t(t − 1)(t − 2)(t − 3) ≡ −10 mod t^{2} + 1 a similar computation as in Example 6.4 yields that 𝔅_{S}(∂(ξ)) = 0.

By Theorem 5.9, assuming that deg(ρ) = 6 and ρ_{∞} = 0, the vanishing of diman 𝔅_{S}(ρ) implies that ρ = ∂(σ_{1} + σ_{2}) for two symbols σ_{1} and σ_{2} in k_{2}E(t). The following example shows that the converse to this statement does not hold.

6.6. Example. Consider ξ = {(t^{2} − 1)(t^{2} − 2), −1} + {t(t − 3), 11} in k_{2}ℚ_{11}(t). Note that S = Supp(∂(ξ)) consists of the factors of (t^{2} − 1)(t^{2} − 2)t(t − 3) and thus has degree 6. A computation shows that 𝔅_{S}(∂(ξ)) = [⟨1, 1, 11, 11⟩] ≠ 0 in W ℚ_{11}. Hence diman 𝔅_{S}(∂(ξ)) = 4.

Recall that B_{S}(ρ) lies in the fundamental ideal IE if and only if deg(S) is even. The following example illustrates that, in general, for finite subsets S_{1} and S_{2} of P with deg(S_{1}) ≡ deg(S_{2}) mod 2 and Supp(ρ) ⊆ S_{1} ∩ S_{2}, we may have that B_{S_{1}}(ρ) ≠ B_{S_{2}}(ρ).

6.7. Example. Consider ξ = {t^{4} − 1, −1} + {t^{2} − t − 1, 3} ∈ k_{2}ℚ_{3}(t) and set ρ = ∂(ξ) and S = Supp(ρ). We have ρ_{∞} = 0, and since −1 is a square in the residue field of t^{2} + 1, we also have ρ_{t^{2}+1}(ξ) = 0. It follows that S consists of the factors of (t^{2} − 1)(t^{2} − t − 1). Hence deg(S) = 4, and a computation yields that B_{S}(ρ) = [⟨1, 1, 3, 3⟩] ≠ 0 in k_{2}ℚ_{3}. In particular, ρ is not represented by a symbol, by Theorem 5.12. Another computation based on Theorem 3.8 further shows that

𝔅 (\frac{t^{2} − t − 1}{t^{4} − 1}) = 𝔅 (\frac{t^{4} − 1}{t^{2} − t − 1}) = 0 in k_{2}ℚ_{3} .

For S′ = S ∪ {t^{2} + 1} we obtain that Supp(ρ) = S ⊆ S′, deg(S′) = 6 and P_{S′} = (t^{4} − 1)(t^{2} − t − 1). Since t^{4} − 1 and t^{2} − t − 1 are coprime, we conclude by Proposition 3.5 that

B_{S′}(ρ) = 𝔅 (\frac{−1(t^{2} − t − 1)}{t^{4} − 1}) + 𝔅 (\frac{3(t^{4} − 1)}{t^{2} − t − 1}) = 0 ≠ B_{S}(ρ) .

6.8. Question. Assume that deg(ρ) = 6 and ρ_{∞} = 0. Set S = Supp(ρ). Does any of the following two properties imply the other one?

(i) ρ = ∂(σ_{1} + σ_{2}) for two symbols σ_{1}, σ_{2} ∈ k_{2}E(t). (ii) diman 𝔅_{S}(ρ) ⩽ 4.

Acknowledgments. This work was supported by the FWO Odysseus Programme (project Explicit Methods in Quadratic Form Theory), funded by the Fonds Wetenschappelijk Onderzoek – Vlaanderen. We would further like to express our gratitude to Jean-Pierre Tignol and to Chris Blondia for their supportive influence on this work.

References

[1] K.J. Becher, M. Raczek. On the second K-group of a rational function field. Pacific J. Math. 262 (2013): 1–9.

[2] P. Gille and T. Szamuely. Central simple algebras and Galois cohomology. Cambridge University Press (2006).

[3] G. Heinig, K. Rost. Introduction to Bezoutians. Numerical methods for structured matrices and applications, 25–118, Oper. Theory Adv. Appl., 199, Birkhäuser Verlag, Basel, 2010.

[4] B.È. Kunyavskiǐ, L.H. Rowen, S.V. Tikhonov, V.I. Yanchevskiǐ. Bicyclic algebras of prime exponent over function fields. Trans. Amer. Math. Soc. 358 (2006): 2579–2610.

[5] T.Y. Lam. Introduction to quadratic forms over fields. Graduate Studies in Mathematics, 67, American Mathematical Society, Providence, RI, 2005.

[6] H.W. Lenstra Jr., K_{2} of a global field consists of symbols. Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), pp. 6973. Lecture Notes in Math., Vol. 551, Springer, Berlin, 1976.

[7] J. Milnor. Algebraic K-theory and quadratic forms. Invent. Math. 9 (1970): 318–344.

[8] A. S. Sivatski. Faddeev invariants for central simple algebras over rational function fields. Manuscripta Math. 145 (2014): 71–88.

Universiteit Antwerpen, Departement Wiskunde–Informatica, Middelheimlaan 1, 2020 Antwerpen, Belgium.

  E-mail address: karimjohannes.becher@uantwerpen.be, melanie.raczek@telenet.be