SPRINGER’S THEOREM FOR TAME QUADRATIC FORMS OVER HENSELIAN FIELDS

M. A. ELOMARY AND J.-P. TIGNOL

Abstract. A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tamely ramified extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation, hence also isomorphic to a direct sum of copies of the Witt group of the residue field indexed by the value group modulo 2.

A celebrated theorem of Springer [8] establishes an isomorphism between the Witt group of a complete discretely valued field and the direct sum of two copies of the Witt group of the residue field, provided the residue characteristic is different from 2. Springer also considered the case where the residue characteristic is 2, and he pointed out the extra complications that arise from the fact that the residue forms are not necessarily nonsingular, even when the residue field is perfect. The exhaustive analysis by Aravire–Jacob [1] shows that the description of the Witt group of a dyadic Henselian field is extremely delicate, even when the field is maximally complete of characteristic 2.

Our purpose is to show that, by contrast, Springer’s original techniques—as revisited in [7]—yield a very general characteristic-free version of Springer’s theorem for the tame part of the Witt group. Our main result is the following: let F be a field with a Henselian valuation v, with value group Γ_{F} and residue field \overline{F} of arbitrary characteristic; there is a group isomorphism

∂: I_{qt}(F) →^{∼} ⨁_{Γ_{F} /2Γ_{F}} I_{q}(\overline{F})

where I_{q}(\overline{F}) is the quadratic Witt group of \overline{F}, consisting of Witt-equivalence classes of even-dimensional nonsingular quadratic forms over \overline{F}, and I_{qt}(F) ⊆ I_{q}(F) is the subgroup of Witt classes that are split by a tamely ramified extension of F. The isomorphism ∂ is defined by residue forms, which depend on the choice of “uniformizing parameters” π_{δ} ∈ F^{×} for representatives δ of the various cosets of Γ_{F} modulo 2Γ_{F}. It is actually obtained as the composition of a canonical isomorphism

(1)  I_{qt}(F) →^{∼} I_{q}(gr(F)),

2000 Mathematics Subject Classification. 11E81.

The first author would like to thank the second author and UCL for their hospitality while the work for this paper was done. He gratefully acknowledges support from the UCL Conseil des relations internationales. The second author is partially supported by the F.R.S.–FNRS (Belgium).

where gr(F) is the graded ring associated with the filtration of F defined by the valuation v, and an isomorphism

(2)  I_{q}(gr(F)) →^{∼} ⨁_{Γ_{F} /2Γ_{F}} I_{q}(\overline{F})

depending on the choice of uniformizing parameters. If char \overline{F} ≠ 2, the isomorphism ∂ extends to the whole Witt group W (F), as is well-known (see [10, Satz 3.1] of [7], for instance). The case where char \overline{F} = 2 was also considered by Tietze in [10]^{1}, who also obtained an isomorphism from a subgroup U(F) ⊆ I_{q}(F) onto ⨁_{Γ_{F} /2Γ_{F}} I_{q}(\overline{F}). Tietze’s method is quite different: it uses a presentation of Witt groups by generators and relations, and U(F) is defined in [10] as the subgroup generated by the Witt classes of quadratic forms ax^{2} +bxy +cy^{2} with v(a) = v(b) = v(c). Comparing Tietze’s results with ours, it appears that actually U(F) = I_{qt}(F), see Corollary 17.

Sections 1 and 2 introduce the formalism of norms on vector spaces and graded rings. The Witt group of graded fields is discussed in §1, where we prove the isomorphism (2): see Theorem 2. The isomorphism (1) is obtained in §2 (see Corollary 11), where I_{qt}(F) is defined as the group of Witt classes of quadratic spaces that admit a certain type of norms, which we call tame norms. In §3, we establish the alternative characterization of I_{qt}(F) in terms of tame splitting fields.

The following notation will be used consistently throughout: F is a field with a (Krull) valuation v: F → Γ∪{∞}, where Γ is a totally ordered group. Without loss of generality, we assume Γ is divisible, since we may substitute for Γ its divisible hull. We let Γ_{F} denote the value group of F, i.e., Γ_{F} = v(F^{×}) ⊆ Γ, and let \overline{F} denote the residue field of F.

For quadratic forms, we generally follow the terminology of [3]: if q: V → F is a quadratic form on an F -vector space V, we let b: V × V → F denote the polar form

b(x, y) = q(x + y) − q(x) − q(y) for x, y ∈ V.

The form b is nondegenerate if x = 0 is the only vector such that b(x, y) = 0 for all y ∈ V. We call the quadratic form q regular if x = 0 is the only vector in V such that q(x) = 0 and b(x, y) = 0 for all y ∈ V, and nondegenerate if it remains regular after scalar extension to an algebraic closure of F. We say q is nonsingular if its polar form is nondegenerate; thus, if char F ≠ 2 or if dim q is even, q is nonsingular if and only if it is nondegenerate. If char F = 2, all nonsingular forms are evendimensional since their polar form is nondegenerate and alternating, but x_{1}x_{2} + x_{3}^{2} is an odd-dimensional nondegenerate form.

1. Quadratic Witt group of graded fields

Let Γ be a divisible torsion-free abelian group, which will contain the degrees of all the graded structures we shall consider. Graded commutative rings in which every nonzero homogeneous element is invertible are called graded fields, and graded modules over graded fields are called graded vector spaces. Since Γ is torsion-free, graded fields are domains and graded vector spaces are free modules, see [5, §1].

1We are grateful to S. Garibaldi for calling our attention on this reference.

The rank of a graded vector space is called its dimension. Let

F = ⨁_{γ∈Γ} F_{γ}

be a graded field and

V = ⨁_{γ∈Γ} V_{γ}

be a graded F-vector space. We let Γ_{F}, Γ_{V} denote the sets of degrees of F and V, i.e.,

Γ_{F} = {γ ∈ Γ | F_{γ} ≠ {0}}, Γ_{V} = {γ ∈ Γ | V_{γ} ≠ {0}}.

The set Γ_{F} is a subgroup of Γ, and Γ_{V} is a union of cosets of Γ_{F}. For each coset Λ ∈ Γ/Γ_{F}, let

V_{Λ} = ⨁_{λ∈Λ} V_{λ}

(so V_{Λ} = {0} if Λ ⊄ Γ_{V}). If Γ_{V} is the disjoint union of cosets Λ_{1}, …, Λ_{n} ∈ Γ/Γ_{F}, i.e., Γ_{V} = Λ_{1} ⊔ … ⊔ Λ_{n}, we have a canonical decomposition of V into graded sub-vector spaces

(3)  V = ⨁_{i=1}^{n} V_{Λ_{i}}.

Note that the homogeneous component F_{0} of F is a field, and each V_{γ} for γ ∈ Γ is an F_{0}-vector space. Picking an element λ_{i} ∈ Λ_{i} for each i = 1, …, n, we have

dim_{F} V = ∑_{i=1}^{n} dim_{F} V_{Λ_{i}} = ∑_{i=1}^{n} dim_{F_{0}} V_{λ_{i}}.

Let V be a finite-dimensional graded vector space over a graded field F. A graded quadratic form on V is a map

q: V → F

satisfying the following conditions involving q and its polar form b: V × V → F defined by

b(x, y) = q(x + y) − q(x) − q(y) :

(i) q(xa) = q(x)a^{2} for all x ∈ V, a ∈ F; (ii) b is an F-bilinear form on V; (iii) q(V_{γ}) ⊆ F_{2γ} for all γ ∈ Γ; (iv) b(V_{γ}, V_{δ}) ⊆ F_{γ+δ} for all γ, δ ∈ Γ.

These conditions imply in particular that

(4)  q(V_{γ}) = \{0\} if γ ∉ \frac{1}{2}Γ_{F} and b(V_{γ}, V_{δ}) = \{0\} if γ + δ ∉ Γ_{F}.

The graded quadratic form q is called nonsingular if its polar form b is nondegenerate, i.e., x = 0 is the only vector such that b(x, y) = 0 for all y ∈ V. It is called hyperbolic if it is nonsingular and V = U ⊕ W for some graded subspaces U, W such that q(U) = q(W) = {0}.

Proposition 1. Let q be a nonsingular graded quadratic form on V. The canonical decomposition (3) yields an orthogonal decomposition

V = ( ⨁_{Λ∈ \frac{1}{2} Γ_{F}/Γ_{F}}^{⊥} V_{Λ}) ⊕^{⊥} W where W = ⨁_{Λ∉ \frac{1}{2} Γ_{F}/Γ_{F}} V_{Λ}.

For each Λ ∈ \frac{1}{2}Γ_{F}/Γ_{F}, the restriction of q to V_{Λ} is nonsingular. Moreover, the restriction of q to W is hyperbolic.

Proof. All assertions except the last one really concern the polar form b; their easy proof (based on the observation (4)) can be found in [7, Proposition 1.1]. It is also proven there that the restriction of b to W is hyperbolic. Since moreover q(V_{Λ}) = \{0\} for all cosets Λ ⊄ \frac{1}{2}Γ_{F} by (4), the proposition follows. □

For each coset Δ ∈ Γ_{F}/2Γ_{F}, fix an element δ ∈ Δ and a nonzero homogeneous element π_{δ} ∈ F_{δ} (δ = 0 and π_{0} = 1 for Δ = 2Γ_{F}) and let

(5)  q_{Δ} : V_{\frac{1}{2} δ} → F_{0},  x ↦ π_{δ}^{−1}q(x).

Since the restriction of q to V_{\frac{1}{2} Δ} is nonsingular, it follows that q_{Δ} is a nonsingular quadratic form on the F_{0}-vector space V_{\frac{1}{2} δ}. Of course, this quadratic form depends on the choice of π_{δ}, except for Δ = 2Γ_{F}.

Mimicking the usual construction, we may define a Witt equivalence of nonsingular graded quadratic forms over a given graded field F and endow the set of Witt-equivalence classes with a group structure using the orthogonal sum. We let I_{q}(F) denote the quadratic Witt group of F, consisting of Witt-equivalence classes of even-dimensional nonsingular graded quadratic forms over F.

Theorem 2. The map that carries each nonsingular graded quadratic form q to the collection of nonsingular F_{0}-quadratic forms (q_{Δ})_{Δ∈Γ_{F}/2Γ_{F}} defines a group isomorphism:

I_{q}(F) →^{∼} ⨁_{Γ_{F}/2Γ_{F}} I_{q}(F_{0}).

This isomorphism depends on the choice of the homogeneous elements π_{δ} ∈ F_{δ}.

The proof is routine. See [7, Proposition 1.5(iv)] for the (more complicated) case of even hermitian forms over graded division algebras with involution.

2. Norms and residues

Let (F, v) be an arbitrary valued field and V a finite-dimensional F -vector space. We recall from [7] and [11] that a v-value function on V is a map

α: V → Γ ∪ {∞}

satisfying the following properties, for x, y ∈ V and λ ∈ F:

(i) α(x) = ∞ if and only if x = 0; (ii) α(xλ) = α(x) + v(λ); (iii) α(x + y) ≥ min(α(x), α(y)).

The v-value function α is called a v-norm if there is a base (e_{i})_{i=1}^{n} of V that splits α in the following sense:

α(∑_{i=1}^{n} e_{i}λ_{i}) = min(α(e_{i}λ_{i}) | i = 1, …, n) for λ_{1}, …, λ_{n} ∈ F.

Any v-value function α on V defines a filtration of V by modules over the valuation ring of F: for any γ ∈ Γ we let

V^{≥γ} = {x ∈ V | α(x) ≥ γ}, V^{>γ} = {x ∈ V | α(x) > γ}, V_{γ} = V^{≥γ}/V^{>γ}, and we define

gr_{α}(V) = ⨁_{γ∈Γ} V_{γ}.

Similarly, let gr(F) be the graded ring associated to the filtration of F defined by the valuation. Since every nonzero homogeneous element of gr(F) is invertible, this ring is a graded field. The F -vector space structure on V induces on gr_{α}(V) a structure of graded gr(F)-module, so gr_{α}(V) is a graded gr(F)-vector space. For any nonzero x ∈ V we let

x̃ = x + V^{>α(x)} ∈ V_{α(x)} ⊆ gr_{α}(V).

We also let ˜0 = 0 and use a similar notation for elements in gr(F). It is shown in [7, Corollary 2.3] that a base (e_{i})_{i=1}^{n} of V splits α if and only if (ẽ_{i})_{i=1}^{n} is a gr(F)-base of gr_{α}(V), and that α is a norm if and only if

dim_{gr(F)} gr_{α}(V) = dim_{F} V.

Now, let q: V → F be an arbitrary quadratic form, with polar form b: V × V → F. We say that a v-value function α is bounded by q, and write α ≺ q, if the following two conditions hold:

(a) v(b(x, y)) ≥ α(x) + α(y) for all x, y ∈ V, and

(b) v(q(x)) ≥ 2α(x) for all x ∈ V.

Of course, letting x = y in (a) yields (b) if char F ≠ 2. On the other hand, (b) does not imply (a): see Example 3 below. Note that Springer in [9] requires only (b), whereas Goldman–Iwahori in [4] require only (a). Both conditions are required by Bruhat–Tits in [2, Définition 2.1].

For each γ ∈ Γ, let p_{γ}: F^{≥γ} → F_{γ} be the canonical map. When α ≺ q we define maps

q̃_{α}: V_{γ} → F_{2γ} and b̃_{α}: V_{γ} × V_{δ} → F_{γ+δ} for γ, δ ∈ Γ

by

q̃_{α}(x̃) = p_{2γ}(q(x)) and b̃_{α}(x̃, ỹ) = p_{γ+δ}(b(x, y))

for x, y ∈ V with α(x) = γ and α(y) = δ. We extend b̃_{α} to gr_{α}(V) by bilinearity and define

q̃_{α}: gr_{α}(V) → gr(F)

as follows: for x̃_{γ} ∈ V_{γ},

q̃_{α}(∑_{γ∈Γ} x̃_{γ}) = ∑_{γ} q̃_{α}(x̃_{γ}) + ∑_{γ<δ} b̃_{α}(x̃_{γ}, x̃_{δ}).

The map q̃_{α} is a quadratic form on gr_{α}(V) with polar form b̃_{α}. It is a graded quadratic form since q̃_{α}(V_{γ}) ⊆ F_{2γ} and b̃_{α}(V_{γ}, V_{δ}) ⊆ F_{γ+δ} for γ, δ ∈ Γ. The straightforward verifications are omitted.

Example 3. Suppose v is a discrete valuation on F with Γ_{F} = ℤ and let V = F^{⊕2} with the hyperbolic quadratic form q(x_{1}, x_{2}) = x_{1}x_{2}. Define a norm α: V → \frac{1}{2}ℤ ∪ \{∞\} by

α(x_{1}, x_{2}) = min(v(x_{1}), v(x_{2}) + \frac{1}{2}) for x_{1}, x_{2} ∈ F.

Clearly, we have v(q(x)) ≥ 2α(x) for all x ∈ V; but b((1, 0), (0, 1)) = 1, so

v(b((1, 0), (0, 1))) = 0 < α(1, 0) + α(0, 1) = \frac{1}{2}.

Thus, condition (b) holds but not (a). Also, note that for x = (1, 0) and x′ = (1, 1) we have x̃ = x′ in gr_{α}(V), but q(x) = 0 and q(x′) = 1, so q̃_{α}(x̃) is not well-defined.

By contrast, the map β: V → ℤ ∪ {∞} defined by

β(x_{1}, x_{2}) = min(v(x_{1}), v(x_{2})) for x_{1}, x_{2} ∈ F

is bounded by q. The induced quadratic form q̃_{β} is hyperbolic.

Example 4. Let (F, v) be an arbitrary valued field and let K/F be an arbitrary quadratic extension. Let N: K → F be the norm form, which is a quadratic form. We consider two cases: Case 1: The valuation v extends to two different valuations v_{1}, v_{2} on K.

Then K/F is not purely inseparable, hence it is a Galois extension. Let σ: K → K be the nontrivial automorphism of K/F, so v_{2} = v_{1} ∘ σ. We have v(N(x)) = v_{1}(x) + v_{2}(x) for all x ∈ V. The map α: K → Γ ∪ {∞} defined by

α(x) = min(v_{1}(x), v_{2}(x)) for x ∈ K

is a v-norm on K by [11, Corollary 1.7], and it is readily checked that α ≺ N. Let u ∈ K be such that v_{1}(u) ≠ v_{2}(u). Then σ(u) ≠ −u, hence after scaling by a suitable factor in F^{×} we may assume u+σ(u) = 1. Since v_{1}(σ(u)) = v_{2}(u) ≠ v_{1}(u), this equation yields

0 = min(v_{1}(u), v_{1}(σ(u))) = α(u).

It follows that b̃_{α}(˜1, ũ) = ˜1, so the form Ñ_{α} is nonsingular. Moreover, Ñ_{α}(ũ) = 0, so Ñ_{α} is hyperbolic. Case 2: The valuation v has a unique extension to K.

We again write v for the valuation on K extending v. This valuation is a v-value function on K and it is readily checked that v ≺ N; in fact v(N(x)) = 2v(x) for all x ∈ K. If K/F is immediate (i.e., \overline{K} = \overline{F} and Γ_{K} = Γ_{F}), then gr_{v}(K) = gr(F), so v is not a norm. Otherwise, gr_{v}(K) is a quadratic extension of gr(F), and Ñ_{α} is the norm form of that extension. It is nonsingular if and only if K/F is tame, i.e., either K/F is totally ramified and char \overline{F} ≠ 2, or \overline{K}/\overline{F} is a separable quadratic extension (i.e., K/F is inertial).

This last example suggests the following definition:

Definition 5. Let (F, v) be an arbitrary valued field and (V, q) be a quadratic space over F. A v-norm α on V is called a tame norm compatible with q if α ≺ q and the induced quadratic form q̃_{α} on gr_{α}(V) is nonsingular.

For any quadratic space (V, q) and any v-value function α on V such that α ≺ q, it is clear that b̃_{α}(x̃, η) = 0 for all η ∈ gr_{α}(V) if b(x, y) = 0 for all y ∈ V. Therefore, tame compatible norms exist only for nonsingular forms. If char \overline{F} ≠ 2, the tame norms compatible with a quadratic form are exactly the norms that are compatible with its polar form, in the terminology of [7]; see [7, Remark 3.2]. In that case, for every nonsingular quadratic form there is a tame compatible norm, see [7, Corollary 3.6]. If char \overline{F} = 2, the tame norms compatible with a quadratic form q are those that are compatible with its polar form and that moreover satisfy condition (b): v(q(x)) ≥ 2α(x) for all x ∈ V. There are nonsingular forms for which there is no tame compatible norm, for instance the norm forms of totally ramified quadratic extensions of Henselian dyadic fields: see Theorem 18.

Lemma 6. Let (V_{1}, q_{1}) and (V_{2}, q_{2}) be quadratic spaces over F, and let α_{1}, α_{2} be tame v-norms on V_{1}, V_{2} that are compatible with q_{1} and q_{2} respectively. Define α_{1} ⊕ α_{2} : V_{1} ⊕ V_{2} → Γ ∪ {∞} by

(α_{1} ⊕ α_{2})(x_{1}, x_{2}) = min(α_{1}(x_{1}), α_{2}(x_{2})) for x_{1} ∈ V_{1} and x_{2} ∈ V_{2}.

Then α_{1} ⊕ α_{2} is a tame v-norm on V_{1} ⊕ V_{2} compatible with q_{1} ⊥ q_{2}, and there is a canonical identification of graded quadratic spaces

(gr_{α_{1}⊕α_{2}}(V_{1} ⊕ V_{2}), { \atop (q_{1} ⊥ q_{2})}_{α_{1}⊕α_{2}}) = (gr_{α_{1}}(V_{1}), q̃_{1}_{α_{1}}) ⊥ (gr_{α_{2}}(V_{2}), q̃_{2}_{α_{2}}).

The easy proof is omitted; see [7, Example 3.7(iii)].

Lemma 7. Let (V, q) be a quadratic space over F and let α be a tame v-norm on V compatible with q. The Witt indices of q and q̃_{α} are related by

𝔦_{0}(q) ≤ 𝔦_{0}(q̃_{α}).

In particular, if q is hyperbolic, then q̃_{α} is hyperbolic.

Proof. If U ⊆ V is a totally q-isotropic subspace, then gr_{α}(U) ⊆ gr_{α}(V) is a totally q̃_{α}-isotropic subspace with dim gr_{α}(U) = dim U. □

Let I_{q}(F) denote the quadratic Witt group of F, consisting of Witt-equivalence classes of even-dimensional nonsingular quadratic forms; see [3, §8.B]. From Lemma 6, it follows that the Witt-equivalence classes of even-dimensional nonsingular quadratic forms that admit a tame compatible norm form a subgroup of I_{q}(F). We let I_{qt}(F) denote this subgroup, which we call the tame quadratic Witt group of F.

Proposition 8. There is a well-defined group epimorphism

∂: I_{qt}(F) → I_{q}(gr(F))

that carries the Witt class of any even-dimensional nonsingular quadratic form q with a tame compatible norm α to the Witt class of q̃_{α}.

Proof. If char \overline{F} ≠ 2, this is shown in [7, Theorem 3.11] (and holds for odddimensional nondegenerate quadratic forms as well). The arguments also hold if char \overline{F} = 2: to see that the Witt equivalence class of q̃_{α} does not depend on the choice of the tame compatible norm α, suppose β is another tame compatible norm. Then α ⊕ β is a tame norm compatible with the hyperbolic form q ⊥ −q, hence Lemmas 6 and 7 show that q̃_{α} ⊥ −q̃_{β} is hyperbolic.

To prove surjectivity of ∂ when char \overline{F} = 2, it suffices to show that the Witt class of every nonsingular binary form over gr(F) is in the image. Let V be a 2dimensional graded vector space over gr(F), and let φ: V → gr(F) be a nonsingular quadratic form. Since the corresponding polar form b_{φ} is nonsingular, we may find homogeneous vectors ε_{1}, ε_{2} ∈ V such that b_{φ}(ε_{1}, ε_{2}) = 1. These vectors form a base of V. If φ(ε_{1}) = 0 or φ(ε_{2}) = 0, then φ is hyperbolic hence its Witt class is zero, which lies in the image of ∂. If φ(ε_{1}), φ(ε_{2}) are nonzero, then we may find a_{1}, a_{2} ∈ F^{×} such that φ(ε_{1}) = ã_{1} and φ(ε_{2}) = ã_{2}. Note that the condition b_{φ}(ε_{1}, ε_{2}) = 1 implies that deg ε_{1} + deg ε_{2} = 0, hence v(a_{1}) + v(a_{2}) = 0. Consider a 2-dimensional F -vector space U with base e_{1}, e_{2} and quadratic form

q(e_{1}x_{1} + e_{2}x_{2}) = a_{1}x_{1}^{2} + x_{1}x_{2} + a_{2}x_{2}^{2} for x_{1}, x_{2} ∈ F.

Straightforward computations show that q is a nonsingular quadratic form and that the map α: U → Γ ∪ {∞} defined by

α(e_{1}x_{1} + e_{2}x_{2}) = min(\frac{1}{2}v(a_{1}) + v(x_{1}), \frac{1}{2}v(a_{2}) + v(x_{2})) for x_{1}, x_{2} ∈ F

is a tame norm compatible with q, such that (gr_{α}(U), q̃_{α}) ≃ (V, φ) under the map e_{1}x_{1} + e_{2}x_{2} ↦ ε_{1}x̃_{1} + ε_{2}x̃_{2}. Thus, the Witt class of φ is in the image of ∂. □

Our next goal is to show that ∂ is an isomorphism when the valuation v on F is Henselian.

Lemma 9. Let (V, q) be a quadratic space over F. Suppose v is Henselian and x, y ∈ V are such that

2v(b(x, y)) < v(q(x)) + v(q(y)).

Then there is a q-isotropic vector in the span of x and y.

Proof. This was observed by several authors, in particular Springer [8, Proposition 1] (see also [10, Lemma (2.2)]). We give a proof for completeness. First, note that if z_{1}, z_{2} ∈ V are multiple of each other, then

(6)  2v(b(z_{1}, z_{2})) = v(q(z_{1})) + v(q(z_{2})) + 2v(2) ≥ v(q(z_{1})) + v(q(z_{2})).

In particular, under the hypotheses of the lemma, x and y are not multiple of each other. Set z = yq(x)b(x, y)^{−1}, so b(x, z) = q(x). For all λ ∈ F we have

(7)  q(xλ + z) = q(x)(λ^{2} + λ + q(x)q(y)b(x, y)^{−2}).

The second factor on the right side is a polynomial P (λ) with coefficients in the valuation ring of F. Its image in \overline{F} [λ] is λ(λ + 1). By Hensel’s Lemma, P (λ) has a root λ_{0} ∈ F. The vector xλ_{0} + z is nonzero since x and y are not multiple of each other, and it is an isotropic vector of q. □

Theorem 10. Let (V, q) be a quadratic space over F. Assume the valuation v on F is Henselian. Suppose α: V → Γ ∪ {∞} is a tame norm compatible with q, and consider the induced quadratic form q̃_{α} on gr_{α}(V). The forms q and q̃_{α} have the same Witt index:

𝔦_{0}(q) = 𝔦_{0}(q̃_{α}).

Proof. Lemma 7 already yields 𝔦_{0}(q) ≤ 𝔦_{0}(q̃_{α}). To prove the reverse inequality, we argue by induction on dim V, and copy the proof of [7, Proposition 4.3]. If q̃_{α} is isotropic, we may find a homogeneous isotropic vector x̃ by taking the homogeneous component of smallest degree of an arbitrary isotropic vector. Since α is a tame norm compatible with q, the polar form b̃_{α} is nondegenerate. Therefore, we may find a homogeneous vector ỹ such that b̃_{α}(x̃, ỹ) ≠ 0. Let W ⊆ V be the subspace spanned by x and y, so gr_{α}(W) ⊆ gr_{α}(V) is the graded subspace spanned by x̃ and ỹ. We claim that W is a hyperbolic plane. To see this, observe that the equation q̃_{α}(x̃) = 0 yields v(q(x)) > 2α(x), while b̃_{α}(x̃, ỹ) ≠ 0 shows that v(b(x, y)) = α(x) + α(y). Since v(q(y)) ≥ 2α(y), it follows that 2v(b(x, y)) < v(q(x)) + v(q(y)). Therefore, Lemma 9 shows that W contains a q-isotropic vector. The restriction of q to W is nonsingular since b(x, y) ≠ 0, hence W is a hyperbolic plane.

Let q′ denote the restriction of q to W^{⊥}, so 𝔦_{0}(q) = 1 + 𝔦_{0}(q′). By the choice of x and y, the bilinear form b̃_{α} is nondegenerate on gr_{α}(W). Therefore, the norm α|_{W} is compatible with the restriction of the polar form b|_{W}, and [7, Proposition 3.8] shows that W^{⊥} is a splitting complement of W with respect to α, i.e., α = α|_{W} ⊕ α|_{W^{⊥}}. Thus,

gr_{α}(V) = gr_{α}(W) ⊥ gr_{α}(W^{⊥}),

hence 𝔦_{0}(q̃_{α}) = 1 + 𝔦_{0}(q′_{α}) since gr_{α}(W) is a hyperbolic plane. The induction hypothesis yields 𝔦_{0}(q′) ≥ 𝔦_{0}(q′_{α}), hence 𝔦_{0}(q) ≥ 𝔦_{0}(q̃_{α}) and the proof is complete. □

Corollary 11. The homomorphism ∂ of Proposition 8 is an isomorphism if F is Henselian.

Proof. Surjectivity of ∂ was shown in Proposition 8, and injectivity follows from Theorem 10. □

3. Tame quadratic forms over Henselian fields

In this section, we show that the tame quadratic Witt group of a Henselian field is the Witt kernel of the scalar extension map to the maximal tame extension.

Let F be a field with a valuation v: F → Γ ∪ {∞}. Throughout this section, we assume v is Henselian. Let (V, q) be a quadratic space over F with polar form b. If q is anisotropic, we define a map α̂: V → Γ ∪ {∞} by

(8)  α̂(x) = \frac{1}{2}v(q(x))  for x ∈ V.

Proposition 12. The map α̂ is a v-value function and α̂ ≺ q.

Proof. This was proved by Springer [8, Proposition 1] (and attributed to M. Eichler) in the case where the valuation is discrete and F is complete. Springer’s arguments hold without change under the more general hypotheses of this section. We include the proof for the reader’s convenience.

By definition of α̂, we clearly have α̂(xλ) = α̂(x) + v(λ) for x ∈ V and λ ∈ F, and α̂(x) = ∞ if and only if x = 0 because q is anisotropic. It is also clear from the definition that v(q(x)) ≥ 2α̂(x) for all x ∈ V. Thus, it only remains to show

(9)  v(b(x, y)) ≥ α̂(x) + α̂(y)  for x, y ∈ V

and

(10)  α̂(x + y) ≥ min(α̂(x), α̂(y))  for x, y ∈ V.

The inequality (9) readily follows from Lemma 9 since q is anisotropic. Property (10) follows, since (9) implies v(b(x, y)) ≥ min(v(q(x)), v(q(y))) for x, y ∈ V, hence

v(q(x + y)) = v(q(x) + b(x, y) + q(y)) ≥ min(v(q(x)), v(q(y))).
□

Now, consider the following property for an anisotropic quadratic form q over F:

(S) v(b(x, y)) > α̂(x) + α̂(y)  for all nonzero x, y ∈ V such that α̂(x) = α̂(y).

Note that this property cannot hold if char \overline{F} ≠ 2 since for x = y it implies v(2) > 0 (see (6)). By (9), we have v(b(x, y)) > 2 min(α̂(x), α̂(y)) if α̂(x) ≠ α̂(y), hence Property (S) has the following equivalent formulation: for all nonzero x, y ∈ V,

(S’)  v(b(x, y)) > 2 min(α̂(x), α̂(y)) = min(v(q(x)), v(q(y))).

Our goal is to prove that Property (S) characterizes the quadratic forms that remain anisotropic over any inertial extension. Thus, until the end of this section, except for the last theorem, we assume char \overline{F} = 2.

We first consider inertial quadratic extensions K/F. The residue extension \overline{K}/\overline{F} is obtained by adjoining to \overline{F} a root of an irreducible polynomial of the form X^{2} + X + \overline{u} for some \overline{u} ∈ \overline{F}, hence K = F (μ) where μ^{2} + μ + u = 0 for some u ∈ F with v(u) = 0. The Henselian valuation on F has a unique extension to K, for which we also use the notation v. Since 1, \overline{μ} are linearly independent over \overline{F}, we have

(11)  v(a + bμ) = min(v(a), v(b))  for a, b ∈ F.

Lemma 13. An anisotropic quadratic form satisfies (S) if and only if it remains anisotropic over every inertial quadratic extension.

Proof. If (S) does not hold, then we can find nonzero vectors x, y ∈ V such that α̂(x) = α̂(y) and v(b(x, y)) = α̂(x) + α̂(y). In view of (6), the vectors x and y are not multiple of each other. Letting z = yq(x)b(x, y)^{−1} as in the proof of Lemma 9, we see from (7) that q becomes isotropic over F (λ_{0}), where λ_{0} is a root of the polynomial λ^{2} + λ + q(x)q(y)b(x, y)^{−2}. Since v(q(x)q(y)b(x, y)^{−2}) = 0, the field F (λ_{0}) is an inertial extension of F.

Conversely, suppose q becomes isotropic over an inertial quadratic extension K of F. We have K = F (μ) where μ^{2} + μ + u = 0 for some u ∈ F with v(u) = 0. Let x_{0} ⊗ 1 + x_{1} ⊗ μ ∈ V ⊗_{F} K be an isotropic vector of q. We have

(12)  q(x_{0} ⊗ 1 + x_{1} ⊗ μ) = (q(x_{0}) − uq(x_{1})) + (b(x_{0}, x_{1}) − q(x_{1}))μ,

hence

q(x_{0}) = uq(x_{1})  and  b(x_{0}, x_{1}) = q(x_{1}).

Since v(u) = 0, it follows that v(q(x_{0})) = v(q(x_{1})) = v(b(x_{0}, x_{1})), hence (S) does not hold. □

Lemma 14. If an anisotropic quadratic form satisfies (S), then its scalar extension to any inertial quadratic extension also satisfies (S).

Proof. Suppose q is an anisotropic quadratic form over F satisfying (S), and K is an inertial quadratic extension of F. By Lemma 13, we know q remains anisotropic over K. We may therefore extend to V ⊗_{F} K the definition of the v-value function α̂ of (8). Let K = F (μ) where μ^{2} + μ + u = 0 for some u ∈ F with v(u) = 0. For x = x_{0} ⊗ 1 + x_{1} ⊗ μ ∈ V ⊗_{F} K we have by (12) and (11)

α̂(x) = \frac{1}{2} min(v(q(x_{0}) − uq(x_{1})), v(b(x_{0}, x_{1}) − q(x_{1}))).

We claim that

α̂(x) = min(α̂(x_{0}), α̂(x_{1})).

We check this formula case-by-case: if v(q(x_{0})) = v(q(x_{1})), then v(q(x_{0})−uq(x_{1})) ≥ v(q(x_{1})) and, by (S), v(b(x_{0}, x_{1}) − q(x_{1})) = v(q(x_{1})); thus, α̂(x) = α̂(x_{0}) = α̂(x_{1}).

If v(q(x_{0})) > v(q(x_{1})), then v(q(x_{0}) − uq(x_{1})) = v(q(x_{1})) and v(b(x_{0}, x_{1})) > v(q(x_{1})) by (S’), hence α̂(x) = α̂(x_{1}).

Likewise, if v(q(x_{0})) < v(q(x_{1})), then v(b(x_{0}, x_{1})) > v(q(x_{0})), hence α̂(x) = α̂(x_{0}). Thus, the claim is proved.

Now, suppose x = x_{0} ⊗ 1 + x_{1} ⊗ μ and y = y_{0} ⊗ 1 + y_{1} ⊗ μ are nonzero vectors in V ⊗_{F} K such that α̂(x) = α̂(y), and let γ = α̂(x), hence

γ = min(α̂(x_{0}), α̂(x_{1})) = min(α̂(y_{0}), α̂(y_{1})).

We have

b(x, y) = (b(x_{0}, y_{0}) − ub(x_{1}, y_{1})) + (b(x_{0}, y_{1}) + b(x_{1}, y_{0}) − b(x_{1}, y_{1}))μ,

hence by (11)

v(b(x, y)) = min(v(b(x_{0}, y_{0}) − ub(x_{1}, y_{1})), v(b(x_{0}, y_{1}) + b(x_{1}, y_{0}) − b(x_{1}, y_{1}))).

By Property (S’) we have

v(b(x_{i}, y_{j})) > min(α̂(x_{i}), α̂(y_{j})) ≥ 2γ for i, j = 0, 1,

hence

v(b(x_{0}, y_{0}) − ub(x_{1}, y_{1})) > 2γ and v(b(x_{0}, y_{1}) + b(x_{1}, y_{0}) − b(x_{1}, y_{1})) > 2γ.

Therefore, v(b(x, y)) > 2γ, and it follows that Property (S) holds for the extension q_{K} of q to K. □

We now turn to odd-degree extensions. Let L/F be an inertial extension of odd degree d. The Henselian valuation v has a unique extension to L, for which we also use the notation v, and we have L = F (λ) for some λ with v(λ) = 0 such that \overline{L} = \overline{F} (\overline{λ}) and the minimal polynomial of \overline{λ} over \overline{F} has degree d. Every anisotropic quadratic form q: V → F remains anisotropic over L by a theorem of Springer (see [3, Corollary 18.5]), hence we may extend the value function α̂ of (8) to V ⊗_{F} L.

Lemma 15. For x = x_{0} ⊗ 1 + x_{1} ⊗ λ + ⋯ + x_{d−1} ⊗ λ^{d−1} ∈ V ⊗_{F} L, we have

α̂(x) = min(α̂(x_{0}), …, α̂(x_{d−1})).

If q satisfies (S), then its extension q_{L} to L also satisfies (S).

Proof. If x = 0, the formula trivially holds. We may therefore assume x_{0}, …, x_{d−1} are not all zero and set

γ = min(α̂(x_{0}), …, α̂(x_{d−1})).

We have

q_{L}(x) = ∑_{0≤i≤d−1} q(x_{i})λ^{2i} + ∑_{0≤i<j≤d−1} b(x_{i}, x_{j})λ^{i+j}.

Since v(q(x_{i})) ≥ 2γ by definition of γ and

v(b(x_{i}, x_{j})) ≥ α̂(x_{i}) + α̂(x_{j}) ≥ 2γ

by (9), it follows that v(q_{L}(x)) ≥ 2γ, hence α̂(x) ≥ γ. To prove that this inequality is an equality, consider the homogeneous component V_{γ} of the graded vector space grα̂(V) and the quadratic map q̃α̂: V_{γ} → F_{2γ} induced by q, as in §2. This map is anisotropic since v(q(z)) = 2α̂(z) for all z ∈ V, by definition of α̂. Since \overline{L}/\overline{F} is an odd-degree extension, this map remains anisotropic after scalar extension to \overline{L}, by a theorem of Springer (see [3, Corollary 18.5]). Therefore, letting x′_{i} = x_{i} +V^{>γ} ∈ V_{γ} for i = 0, …, d − 1, so that x′_{i} = x̃_{i} ≠ 0 if α̂(x_{i}) = γ and x′_{i} = 0 if α̂(x_{i}) > γ, we have

(q̃α̂)\overline{L}(x′_{0} ⊗ 1 + x′_{1} ⊗ \overline{λ} + ⋯ + x′_{d−1} ⊗ \overline{λ}^{d−1}) ≠ 0.

The left side is the image of q_{L}(x) under the canonical map L^{≥2γ} → L_{2γ}, hence v(q_{L}(x)) = 2γ, and α̂(x) = γ. The formula for α̂(x) is thus established.

Now, let x, y ∈ V ⊗_{F} L be nonzero vectors with α̂(x) = α̂(y). Let γ = α̂(x) = α̂(y) and

x = x_{0} ⊗ 1 + ⋯ + x_{d−1} ⊗ λ^{d−1}, y = y_{0} ⊗ 1 + ⋯ + y_{d−1} ⊗ λ^{d−1}

with x_{0}, …, y_{d−1} ∈ V and

γ = min(α̂(x_{0}), …, α̂(x_{d−1})) = min(α̂(y_{0}), …, α̂(y_{d−1})).

We have

b(x, y) = ∑_{i,j=0}^{d−1} b(x_{i}, y_{j})λ^{i+j}.

If Property (S) (hence also (S’)) holds for q, we have

v(b(x_{i}, y_{j})) > 2 min(α̂(x_{i}), α̂(y_{j})) ≥ 2γ for all i, j,

hence also v(b(x, y)) > 2γ. Thus, Property (S) holds for q_{L}. □

Theorem 16. Suppose F is a field with a Henselian dyadic valuation. An anisotropic quadratic form over F satisfies (S) if and only if it remains anisotropic over every inertial extension of F.

Proof. The “if” part follows from Lemma 13. For the converse, suppose q is an anisotropic form over F satisfying (S), and let M be an inertial extension of F. We have to show that q remains anisotropic over M. Substituting for M its Galois closure, we may assume M is Galois over F. Let L ⊆ M be the subfield fixed under some 2-Sylow subgroup of the Galois group. Then L/F is an odd-degree extension and M/L is a Galois 2-extension, hence there is a sequence of field extensions

L = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{r} = M

with [L_{i} : L_{i−1}] = 2 for i = 1, …, r. By a theorem of Springer (see [3, Corollary 18.5]), q remains anisotropic over L, and its extension q_{L} satisfies (S) by Lemma 15. Therefore, q_{L_{1}} is anisotropic by Lemma 13, and it satisfies (S) by Lemma 14. Arguing iteratively, we see that q_{M} is anisotropic (and satisfies (S)). □

Theorem 16 also yields precise information on the quadratic forms that are split by an inertial extension of the base field.

Corollary 17. Let q be an anisotropic quadratic form over the Henselian dyadic field F. If q becomes hyperbolic over some inertial extension of F, then there exist inertial quadratic extensions K_{1}, …, K_{n} of F and a_{1}, …, a_{n} ∈ F^{×} such that

q ≃ ⟨a_{1}⟩N_{1} ⊥ … ⊥ ⟨a_{n}⟩N_{n}

where N_{i} is the (quadratic) norm form of K_{i}/F.

Proof. Since q becomes hyperbolic over some inertial extension of F, Theorem 16 shows that it does not satisfy Property (S). Therefore, by Lemma 13, q becomes isotropic over some inertial quadratic extension K_{1}/F. By [3, Proposition 34.8], we may then find a_{1} ∈ F^{×} and a quadratic form q_{1} such that

q ≃ ⟨a_{1}⟩N_{1} ⊥ q_{1}.

Since q becomes hyperbolic over some inertial extension of F and N_{1} becomes hyperbolic over the inertial extension K_{1}, the form q_{1} also becomes hyperbolic over some inertial extension of F. The corollary follows by induction on the dimension.

□

If we include the split quadratic F -algebra F × F, with hyperbolic norm form, among the inertial quadratic extensions of F, Witt’s decomposition theorem shows that Corollary 17 also holds for isotropic quadratic forms that become hyperbolic over some inertial extension. It also holds in the nondyadic case, because then Springer’s theorem can be used to show that every anisotropic quadratic form split by an inertial extension becomes isotropic over some inertial quadratic extension. Thus, the same argument as in the proof of Corollary 17 applies.

Our last theorem holds without the hypothesis that char \overline{F} = 2.

Theorem 18. Let F be a field with a Henselian valuation v (with arbitrary residue characteristic), and let (V, q) be an even-dimensional nondegenerate quadratic space over F. The following conditions are equivalent:

(a) q becomes hyperbolic over some tame extension of F;

(b) there exists a tame norm α on V compatible with q.

Proof. If char \overline{F} ≠ 2, both conditions hold for all even-dimensional nondegenerate quadratic spaces: (a) because the quadratic closure of F is tame and (b) by [7, Corollary 3.6]. For the rest of the proof, we may thus assume char \overline{F} = 2. We may also assume q is anisotropic because if (b) holds for an anisotropic form it also holds for every Witt-equivalent form by Lemma 6 since hyperbolic forms admit tame compatible norms (see Example 3).

If (a) holds, then by Corollary 17 we may find a decomposition

q ≃ ⟨a_{1}⟩N_{1} ⊥ … ⊥ ⟨a_{n}⟩N_{n}

for some a_{1}, …, a_{n} ∈ F^{×} and for N_{1}, …, N_{n} the norm forms of some inertial quadratic extensions K_{1}, …, K_{n} of F. Example 4 shows that for each i = 1, …, n the unique valuation on K_{i} extending v is a tame norm compatible with N_{i}, hence also with ⟨a_{i}⟩N_{i}. The direct sum of these norms is a tame norm compatible with q by Lemma 6, hence condition (b) holds.

Conversely, suppose there is a tame norm α on V compatible with q. Then we may find a separable extension L′ of \overline{F} such that q̃_{α} splits after scalar extension to gr(F) ⊗\overline{F} L′. Let L be an inertial lift of L′, i.e., L/F is an inertial extension such that \overline{L} = L′. Write again v for the unique extension of v to L. Then α ⊗ v is a tame norm on V ⊗_{F} L compatible with q_{L}, and (q̃_{L})_{α⊗v} = (q̃_{α})_{gr(L)} is split since gr(L) = gr(F) ⊗\overline{F} L′. Therefore, Theorem 10 shows that q_{L} is hyperbolic. □

As an example, consider the field F = ℚ_{2}((t)) of Laurent series in one indeterminate over the field of 2-adic numbers. The composite of the 2-adic valuation on ℚ_{2} and the t-adic valuation on F is a Henselian valuation v on F with value group ℤ^{2} and residue field 𝔽_{2}. We have I_{q}(𝔽_{2}) ≃ ℤ/2ℤ, and the unique nontrivial Witt class is represented by the norm form of 𝔽_{4}. The unique inertial quadratic extension of F is F (\sqrt{5}), and it follows from Theorem 2, Corollary 11, Corollary 17 and Theorem 18 that

I_{qt}(F) ≃ (ℤ/2ℤ)^{4},

with generators the Witt classes of the following forms: ⟨1, −5⟩, ⟨2⟩⟨1, −5⟩, ⟨t⟩⟨1, −5⟩, ⟨2t⟩⟨1, −5⟩.

By contrast, the full Witt groups W (F), I_{q}(F) may also be determined by using Springer’s theorem for the t-adic valuation, since the Witt group of ℚ_{2} is known (see for instance [6, Ch. VI, Remark 2.31]); we thus get

W (F) ≃ (ℤ/8ℤ)^{2} ⊕ (ℤ/2ℤ)^{4},

with generators the Witt classes of ⟨1⟩, ⟨t⟩, ⟨1, −2⟩, ⟨t⟩⟨1, −2⟩, ⟨1, −5⟩, and ⟨t⟩⟨1, −5⟩. Note that 4⟨1⟩ ≃ ⟨1, −2, −5, 10⟩ over ℚ_{2} (see [6, loc. cit.]), hence ⟨2⟩⟨1, −5⟩ is Wittequivalent to ⟨1, −5⟩ − 4⟨1⟩. Therefore,

I_{q}(F)/I_{qt}(F) ≃ (ℤ/4ℤ) ⊕ (ℤ/2ℤ)^{3}

with generators represented by ⟨1, t⟩, ⟨1, 1⟩, ⟨1, −2⟩, and ⟨t⟩⟨1, −2⟩.

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Moulay Ismail University, F.S.T. Errachidia, B.P. 509, Boutalamine, Errachidia, Morocco

E-mail address: elomary@fste-umi.ac.ma

Université de Louvain, B 1348 Louvain-la-Neuve, Belgium

E-mail address: Jean-Pierre.Tignol@uclouvain.be