On the Pfister Number of Quadratic Forms

R. Parimala, V. Suresh, and J.-P. Tignol

Abstract. The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of −1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n − 2 terms and not less. The number of 2-fold Pfister forms needed to express a quadratic form of dimension 6 with trivial discriminant is determined in various cases.

Introduction

Throughout this paper, k denotes a field of characteristic different from 2 in which −1 is a square. We use the same notation for a quadratic form over k and its Witt equivalence class in the Witt ring W (k). As usual, the quadratic form ∑_{i=1}^{n} a_{i}X^{i} with a_{i} ∈ k^{×} is denoted by ⟨a_{1}, …, a_{n}⟩. Since −1 is a square in k, the form ⟨1, …, 1⟩ is Witt equivalent to ⟨1⟩ or 0 according as its dimension is odd or even, hence W (k) is an algebra over the field 𝔽_{2} with two elements. Let I(k) be the fundamental ideal of W (k), which consists of the Witt equivalence classes of even-dimensional quadratic forms. For any integer m ≥ 1, the m-th power of I(k) is denoted by I^{m}(k). We say a quadratic form is in I^{m}(k) if its Witt equivalence class is in I^{m}(k). It is well-known that for any m ≥ 1 the ideal I^{m}(k) is generated as a group by the classes of m-fold Pfister forms, i.e., quadratic forms of the following type:

⟨⟨a_{1}, …, a_{m}⟩⟩ = ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{m}⟩,

see [7, Prop. X.1.2]. Brosnan, Reichstein, and Vistoli [3] define the m-Pfister number Pf_{m}(q) of a quadratic form q ∈ I^{m}(k) as the least number of terms in a decomposition of its Witt equivalence class into a sum of m-fold Pfister forms. For m, n ≥ 1, the (m, n)-Pfister number Pf_{k}(m, n) is defined as the supremum of the m-Pfister numbers Pf_{m}(q) where q runs over the quadratic forms of dimension n in I^{m}(K), as K varies over field extensions of k. In [3], Pfister numbers are studied in connection with the essential dimension of algebraic groups.

A related invariant was defined by Parimala and Suresh in [9] (see also Kahn’s paper [5]): the length λ_{m}(q) of a quadratic form q ∈ I^{m}(k) is the least integer r for which there exist m-fold Pfister forms π_{1}, …, π_{r} such that q ≡ π_{1} + ⋯ + π_{r} mod I^{m+1}(k). In [9], the length of quadratic forms was studied with reference to the u-invariant of fields and some bounds were given for the length of quadratic forms in I^{m}(k), 1 ≤ m ≤ 3. Clearly, λ_{m}(q) ≤ Pf_{m}(q).

2000 Mathematics Subject Classification. 11E81.

The first author is partially supported by NSF grant DMS-0653382 and the third author is partially supported by the Fund for Scientific Research F.R.S.–FNRS (Belgium).

The following bounds were given in [3] for Pfister numbers of forms in I(k) and I^{2}(k) (see also Proposition 2.1 below):

Proposition ([3, Prop. 14]). Pf_{k}(1, n) ≤ n and Pf_{k}(2, n) ≤ n − 2.

Zinovy Reichstein raised the following question: Is the estimate for the 2-Pfister number in the proposition sharp, i.e., is Pf_{k}(2, n) = n − 2? In this paper we answer Reichstein’s question in the affirmative by showing that the “generic” quadratic form q_{0} of dimension n with trivial discriminant satisfies Pf_{2}(q_{0}) = n − 2, see Theorem 2.2 and Corollary 2.3. Note that for any quadratic form q of dimension n in I^{2}(k) we have λ_{2}(q) ≤ \frac{n−2}{2} (cf. [5, Prop. 1.1]); therefore for the generic form q_{0} the inequality λ_{2}(q_{0}) ≤ Pf_{2}(q_{0}) is strict.

The proof of Theorem 2.2 is easily derived from a discussion of a combinatorial analogue of Pfister numbers in §1. In the last section (§3), which is essentially independent from §§1 and 2, we give some computations of Pfister numbers of quadratic forms of dimension 6.

We are indebted to Zinovy Reichstein for his comments on a first version of this paper, which helped us to improve the wording in several points, and also to Detlev Hoffmann for suggesting an alternative proof of Theorem 2.2. Ideas from this alternative proof were used to simplify our original arguments.

1. A combinatorial analogue

Let V be an arbitrary vector space over the field 𝔽_{2} with 2 elements. We consider the group algebra 𝔽_{2}[V] as a combinatorial analogue of the Witt ring of a field. (Indeed, the Witt ring of any field k of characteristic different from 2 containing a square root of −1 is a homomorphic image of 𝔽_{2}[k^{×}/k^{×2}], see §2.) Since the addition in V is multiplication in 𝔽_{2}[V], it is convenient to denote by X^{v} the image of v ∈ V in 𝔽_{2}[V]; thus

𝔽_{2}[V] = {∑_{v∈V} α_{v}X^{v} | α_{v} ∈ 𝔽_{2} and {v ∈ V | α_{v} ≠ 0} is finite},

with

X^{0} = 1 and X^{u} · X^{v} = X^{u+v} for u, v ∈ V .

We consider the group homomorphisms

ε_{0} : 𝔽_{2}[V] → 𝔽_{2}, ε_{1} : 𝔽_{2}[V] → V

defined by

ε_{0}(∑_{v∈V} α_{v}X^{v}) = ∑_{v∈V} α_{v}, ε_{1}(∑_{v∈V} α_{v}X^{v}) = ∑_{v∈V} α_{v}v.

Thus, ε_{0} is the augmentation map. We denote its kernel by I[V]. It is an ideal since ε_{0} is a ring homomorphism, and it is generated as a group by elements of the form 1 + X^{v} for v ∈ V, which we call 1-fold Pfister elements. For m ≥ 1, the products

(1 + X^{v_{1}}) ⋯ (1 + X^{v_{m}}) ∈ 𝔽_{2}[V]

with v_{1}, …, v_{m} ∈ V are called m-fold Pfister elements. They span the m-th power of I[V], which we denote by I^{m}[V] to mimic the Witt ring notation. Observe that 0 is an m-fold Pfister element for all m, since the product above is 0 if v_{1} = 0.

For ξ = ∑_{v∈V} α_{v}X^{v} ∈ 𝔽_{2}[V] we define the support of ξ by
D(ξ) = {v ∈ V | α_{v} = 1} ⊆ V.

This notation is inspired by the usual notation for the set of represented values of a quadratic form. (See the proof of Theorem 2.2 below for an example of a field E such that W (E) can be identified with a group algebra 𝔽_{2}[V] in such a way that the support of any ξ ∈ 𝔽_{2}[V] is the set of represented values of the corresponding anisotropic quadratic form.)

Lemma 1.1. Let ξ ∈ 𝔽_{2}[V] be a nonzero element, and let d = |D(ξ)| be the cardinality of the support of ξ.

(i) If ξ ∈ I[V], then d ≥ 2 and there are 1-fold Pfister elements π_{1}, …, π_{p}

such that

ξ = π_{1} + ⋯ + π_{p} and p ≤ d.

If moreover 0 ∈ D(ξ), the same property holds with p ≤ d − 1. (ii) If ξ ∈ I[V] and ε_{1}(ξ) = 0, then d ≥ 4 and there exist 2-fold Pfister

elements π_{1}, …, π_{p} such that

ξ = π_{1} + ⋯ + π_{p}  and  p ≤ d − 2.

If moreover 0 ∈ D(ξ), the same property holds with p ≤ d − 3.

Proof. (i) We have d ≠ 0 since ξ ≠ 0, and d is even since ε_{0}(ξ) ≡ d mod 2 and ξ ∈ I[V]. Therefore, we have

ξ = ∑_{v∈D(ξ)} X^{v} = ∑_{v∈D(ξ)}(1 + X^{v}),

proving that ξ is a sum of d terms that are 1-fold Pfister elements. If 0 ∈ D(ξ), one of these terms vanishes since 1 + X^{0} = 0. Thus, (i) is proved.

(ii) Suppose now ξ ∈ I[V] and ε_{1}(ξ) = 0. As in case (i), d is even. If d = 2, the condition ε_{1}(ξ) = 0 yields ξ = 0. Therefore, d ≥ 4. The other assertions are proved by induction on d. Suppose first 0 ∈ D(ξ). Since d ≥ 4 we may find in D(ξ) two distinct nonzero vectors u, v. Define

ξ′ = (1 + X^{u})(1 + X^{v}) + ξ.

We have ξ′ ∈ I[V] and ε_{1}(ξ′) = 0. Moreover,

D(ξ′) ⊆ (D(ξ) \ {0, u, v}) ∪ {u + v},

hence |D(ξ′)| ≤ d − 2. By induction, there exist 2-fold Pfister elements π_{1}, …, π_{p} such that

ξ′ = π_{1} + ⋯ + π_{p} and p ≤ d − 4,

(ξ′ = 0 if d = 4). Then

ξ = π_{1} + ⋯ + π_{p} + (1 + X^{u})(1 + X^{v})

and the number of terms on the right side is at most d − 3. If 0 ∉ D(ξ) we may still define ξ′ as above, and we have

0 ∈ D(ξ′) ⊆ (D(ξ) \ {u, v}) ∪ {0, u + v},

hence |D(ξ′)| ≤ d. The arguments above show that there exist 2-fold Pfister elements π_{1}, …, π_{p} such that

ξ′ = π_{1} + ⋯ + π_{p}  and  p ≤ d − 3.

Then

ξ = π_{1} + ⋯ + π_{p} + (1 + X^{u})(1 + X^{v})

and the number of terms on the right side is at most d − 2. □

Corollary 1.2. I^{2}[V] = {ξ ∈ I[V] | ε_{1}(ξ) = 0}.

Proof. Lemma 1.1(i) shows that I[V] is spanned by 1-fold Pfister elements, hence I^{2}[V] is generated as a group by 2-fold Pfister elements. Since these elements lie in the kernel of ε_{1}, it follows that

I^{2}[V] ⊆ {ξ ∈ I[V] | ε_{1}(ξ) = 0}.

The reverse inclusion readily follows from Lemma 1.1(ii). □

For ξ ∈ I^{m}[V] we define the m-Pfister number Pf_{m}(ξ) as the minimal number of terms in a decomposition of ξ as a sum of m-fold Pfister elements. In particular, Pf_{m}(0) = 0 for all m ≥ 1.

Proposition 1.3. For every ξ ∈ I[V] we have

Pf_{1}(ξ) = \begin{cases}|D(ξ)| & if 0 ∉ D(ξ), \\ |D(ξ)| − 1 & if 0 ∈ D(ξ). \end{cases}

Proof. Let p = Pf_{1}(ξ). Suppose v_{1}, …, v_{p} ∈ V are nonzero vectors such that

ξ = ∑_{i=1}^{p}(1 + X^{v_{i}}) = p + ∑_{i=1}^{p} X^{v_{i}}.

Then D(ξ) ⊆ {0, v_{1}, …, v_{p}}, hence

|D(ξ)| ≤ \begin{cases}p & if 0 ∉ D(ξ), \\ p + 1 & if 0 ∈ D(ξ). \end{cases}

The reverse inequality follows from Lemma 1.1. □

We now turn to 2-Pfister numbers. From Lemma 1.1 it follows that for ξ ≠ 0 in I^{2}[V],

(1)  Pf_{2}(ξ) ≤ |D(ξ)| − 2, and Pf_{2}(ξ) ≤ |D(ξ)| − 3 if 0 ∈ D(ξ).

In the rest of this section, we explicitly construct elements for which the upper bound is reached. The following general observation is crucial for the proof: every linear map φ: V → W between 𝔽_{2}-vector spaces induces a ring homomorphism φ_{∗}: 𝔽_{2}[V] → 𝔽_{2}[W] by

(2)  φ_{∗}(∑_{v∈V} α_{v}X^{v}) = ∑_{v∈V} α_{v}X^{φ(v)}.

The homomorphism φ_{∗} maps 1-fold Pfister elements in 𝔽_{2}[V] to (possibly zero) 1-fold Pfister elements in 𝔽_{2}[W], hence also m-fold Pfister elements in 𝔽_{2}[V] to mfold Pfister elements in 𝔽_{2}[W], for every m ≥ 1. Consequently, for every ξ ∈ I^{m}[V] we have φ_{∗}(ξ) ∈ I^{m}[W] and

Pf_{m}(φ_{∗}(ξ)) ≤ Pf_{m}(ξ).

Now, let V be an 𝔽_{2}-vector space of finite dimension n > 1, and let e = (e_{i})_{i=1}^{n} be a base of V. We define e_{0} = ∑_{i=1}^{n} e_{i} and

ξ_{e} = n + 1 + ∑_{i=0}^{n} X^{e_{i}} ∈ 𝔽_{2}[V].

It is readily verified that ε_{0}(ξ) = 0 and ε_{1}(ξ) = 0, so ξ_{e} ∈ I^{2}[V], and the support of ξ_{e} is

D(ξ_{e}) = \begin{cases}\{0, e_{0}, e_{1}, …, e_{n}\} & if n is even, \\ \{e_{0}, e_{1}, …, e_{n}\} & if n is odd. \end{cases}

Therefore, (1) yields the same inequality when n is odd or even:

(3)  Pf_{2}(ξ_{e}) ≤ n − 1.

The following proposition shows that Pf_{2}(ξ_{e}) reaches the bound in (1).

Proposition 1.4. Pf_{2}(ξ_{e}) = n − 1.

Proof. We use induction on n. If n = 2, we have

ξ_{e} = (1 + X^{e_{1}})(1 + X^{e_{2}}),

so Pf_{2}(ξ_{e}) = 1. If n = 3, then

ξ_{e} = X^{e_{1}} + X^{e_{2}} + X^{e_{3}} + X^{e_{1}}X^{e_{2}}X^{e_{3}}.

This element is not a 2-fold Pfister element since 0 ∉ D(ξ_{e}), hence Pf_{2}(ξ_{e}) > 1. On the other hand, Pf_{2}(ξ_{e}) ≤ 2 by (3), hence Pf_{2}(ξ_{e}) = 2.

For the rest of the proof, suppose n > 3. Let p = Pf_{2}(ξ_{e}) and let π_{1}, …, π_{p} be 2-fold Pfister elements such that

(4)  ξ_{e} = π_{1} + ⋯ + π_{p}.

We have D(ξ_{e}) ⊆ ⋃_{i=1}^{p} D(π_{i}), hence e_{n} ∈ D(π_{i}) for some i = 1, …, p. Renumbering, we may assume e_{n} ∈ D(π_{p}), hence

(5)  π_{p} = 1 + X^{e_{n}} + X^{v} + X^{e_{n}+v}  for some v ∈ V .

Let W ⊆ V be the 𝔽_{2}-span of e_{1}, …, e_{n−1}, and let f_{0} = ∑_{i=1}^{n−1} e_{i} ∈ W. Clearly, f = (e_{i})_{i=1}^{n−1} is a base of W, and the element ξ_{f} ∈ 𝔽_{2}[W] built on the same model as ξ_{e} is

ξ_{f} = n + X^{f_{0}} + ∑_{i=1}^{n−1} X^{e_{i}}.

Consider the linear map φ: V → W defined by

φ(e_{i}) = \begin{cases}e_{i} & for i = 1, …, n − 1, \\ 0 & for i = n. \end{cases}

The ring homomorphism φ_{∗}: 𝔽_{2}[V] → 𝔽_{2}[W] induced by φ as in (2) above satisfies φ_{∗}(X^{e_{n}}) = 1. Since φ(e_{0}) = f_{0}, it follows that φ_{∗}(ξ_{e}) = ξ_{f}, hence (4) yields

ξ_{f} = φ_{∗}(π_{1}) + ⋯ + φ_{∗}(π_{p}).

In view of (5), we have φ_{∗}(π_{p}) = 0, hence the preceding equation shows that Pf_{2}(ξ_{f}) ≤ p − 1. Since dim W = n − 1, the induction hypothesis yields Pf_{2}(ξ_{f}) = n − 2, hence n − 1 ≤ p. The reverse inequality holds by (3), hence the proposition is proved. □

2. Pfister numbers of generic forms

Let k be an arbitrary field of characteristic different from 2 containing a square root of −1, and let V_{k} = k^{×}/k^{×2} be the group of square classes in k, which we view as an 𝔽_{2}-vector space. The map

Ψ: V_{k} → W (k)

defined by Ψ(a k^{×2}) = ⟨a⟩ for a ∈ k^{×} is multiplicative, hence it induces a surjective 𝔽_{2}-algebra homomorphism

Ψ_{∗}: 𝔽_{2}[V_{k}] → W (k).

The map Ψ_{∗} carries 1-fold Pfister elements in 𝔽_{2}[V_{k}] to 1-fold Pfister forms in W (k), hence also m-fold Pfister elements to m-fold Pfister forms for all m ≥ 1. Therefore, Ψ_{∗}(I^{m}[V_{k}]) = I^{m}(k) and we have

(6)  Pf_{m}(Ψ_{∗}(ξ)) ≤ Pf_{m}(ξ)  for all ξ ∈ I^{m}[V_{k}].

We may then use Lemma 1.1 to give a short proof of Proposition 14 of [3], including a minor refinement:

Proposition 2.1. Let q be a quadratic form of dimension n over a field k containing a square root of −1.

(i) If q ∈ I(k), then Pf_{1}(q) ≤ n. If moreover q represents 1, then Pf_{1}(q) ≤ n − 1.

(ii) If q ∈ I^{2}(k), then Pf_{2}(q) ≤ n − 2. If moreover q represents 1, then Pf_{2}(q) ≤ n − 3.

Proof. Let q = ⟨a_{1}, …, a_{n}⟩. Consider then

ξ = (a_{1} k^{×2}) + ⋯ + (a_{n} k^{×2}) ∈ 𝔽_{2}[V_{k}].

We have Ψ_{∗}(ξ) = q and D(ξ) = {a_{1} k^{×2}, …, a_{n} k^{×2}}, so |D(ξ)| ≤ n. If q ∈ I(k), then n is even hence ξ ∈ I[V_{k}]. Lemma 1.1(i) then yields Pf_{1}(ξ) ≤ n, and by (6) it follows that Pf_{1}(q) ≤ n. If q represents 1, then we may assume a_{1} = 1, hence D(ξ) contains the zero element of V_{k}. Lemma 1.1(i) then yields Pf_{1}(ξ) ≤ n − 1, and by (6) it follows that Pf_{1}(q) ≤ n − 1.

If q ∈ I^{2}(k), then a_{1} … a_{n} ∈ k^{×2} hence ε_{1}(ξ) = 0. By Corollary 1.2 we have ξ ∈ I^{2}[V_{k}], and Lemma 1.1(ii) yields Pf_{2}(ξ) ≤ n − 2. Therefore, by (6) we get Pf_{2}(q) ≤ n − 2. Again, if q represents 1 we may assume 0 ∈ D(ξ), and the preceding inequalities can be strengthened to

Pf_{2}(q) ≤ Pf_{2}(ξ) ≤ n − 3.
□

For the rest of this section, fix an arbitrary integer n ≥ 2. Consider n independent indeterminates x_{1}, …, x_{n} over k and let

x_{0} = x_{1} ⋯ x_{n}.

Over the field K = k(x_{1}, …, x_{n}), we consider the following quadratic forms:

q = ⟨x_{1}, …, x_{n}⟩,  q_{0} = ⟨x_{0}, x_{1}, …, x_{n}⟩,
q′ = ⟨1, x_{1}, …, x_{n}⟩,  q′_{0} = ⟨1, x_{0}, x_{1}, …, x_{n}⟩.

If n is even, then q ∈ I(K) and q′_{0} ∈ I^{2}(K). If n is odd, then q′ ∈ I(K) and q_{0} ∈ I^{2}(K).

Theorem 2.2. If n is even, then

Pf_{1}(q) = n  and  Pf_{2}(q′_{0}) = n − 1.

If n is odd, then

Pf_{1}(q′) = n  and  Pf_{2}(q_{0}) = n − 1.

Proof. Let k_{alg} be an algebraic closure of k. Embed k in the field of iterated Laurent series E = k_{alg}((x_{1})) ⋯ ((x_{n})). Applying Springer’s theorem in [7, Cor. VI.1.7] recursively, we obtain a ring isomorphism

Θ: W (E) →^{∼} 𝔽_{2}[(ℤ/2ℤ)^{n}],

which maps W (k_{alg}) onto 𝔽_{2} and maps the quadratic form ⟨x_{i}⟩ to X^{e_{i}}, where e_{i} is the i-th element in the standard base of (ℤ/2ℤ)^{n} as an 𝔽_{2}-vector space, for i = 1, …, n. Note that the (x_{1}, …, x_{n})-adic valuation on E yields an isomorphism

V_{E} = E^{×}/E^{×2} ≃ (ℤ/2ℤ)^{n}

which maps x_{i} E^{×2} to e_{i} for i = 1, …, n. Using this isomorphism as an identification, we may view Θ as the inverse map of Ψ_{∗}: 𝔽_{2}[V_{E}] → W (E), which is thus an isomorphism in this case.

Letting e_{0} = ∑_{i=1}^{n} e_{i}, we have

Θ(q_{E}) = ∑_{i=1}^{n} X^{e_{i}},  Θ(q_{0E}) = ∑_{i=0}^{n} X^{e_{i}},

Θ(q′_{E}) = 1 + ∑_{i=1}^{n} X^{e_{i}},  Θ(q′_{0E}) = 1 + ∑_{i=0}^{n} X^{e_{i}},

hence in the notation of §1 with V = (ℤ/2ℤ)^{n}, we have

ξ_{e} = \begin{cases}Θ(q_{0E}) & if n is odd, \\ Θ(q′_{0E}) & if n is even. \end{cases}

The isomorphism Θ maps m-fold Pfister forms in W (E) to m-fold Pfister elements in 𝔽_{2}[V], hence it preserves m-Pfister numbers. Therefore, Proposition 1.4 yields

Pf_{2}(q_{0E}) = n − 1 if n is odd  and  Pf_{2}(q′_{0E}) = n − 1 if n is even.

Similarly, Proposition 1.3 yields

Pf_{1}(q_{E}) = n if n is even  and  Pf_{1}(q′_{E}) = n if n is odd.

Since scalar extension preserves m-Pfister forms, it follows that

Pf_{1}(q) ≥ n and Pf_{2}(q′_{0}) ≥ n − 1 if n is even,
Pf_{1}(q′) ≥ n and Pf_{2}(q_{0}) ≥ n − 1 if n is odd.

The reverse inequalities follow from Proposition 2.1. □

Corollary 2.3. Pf_{k}(1, m) = m for any even integer m ≥ 2 and Pf_{k}(2, m) = m − 2 for any even integer m ≥ 4.

Proof. For m even, m ≥ 2, the form q above (with n = m) has dimension m and satisfies q ∈ I(F) and Pf_{1}(q) = m, so Pf_{k}(1, m) ≥ m. Similarly, for m even, m ≥ 4, the form q_{0} above (with n = m − 1) has dimension m and satisfies q_{0} ∈ I^{2}(F) and Pf_{2}(q_{0}) = m − 2, so Pf_{k}(2, m) ≥ m − 2. The reverse inequalities follow from [3, Prop. 14] (see the Introduction or Proposition 2.1). □

Remark 2.4. A form with the same 2-Pfister number as q′_{0} can be obtained by scaling q_{0}: we have

⟨x_{1}⟩q_{0} = ⟨1, x_{1}x_{2}, …, x_{1}x_{n}, x_{0}x_{1}⟩

and x_{1}x_{2}, …, x_{1}x_{n} may be regarded as independent indeterminates. If n is odd we have

x_{0}x_{1} ≡ (x_{1}x_{2}) ⋯ (x_{1}x_{n}) mod K^{×2},

hence ⟨x_{1}⟩q_{0} is isometric to a quadratic form like q′_{0} in the indeterminates x_{1}x_{2}, …, x_{1}x_{n}. Embedding K in E as in the proof of Theorem 2.2, we obtain Pf_{2}(⟨x_{1}⟩q_{0}) = n − 2. Details are left to the reader.

3. Low-dimensional forms

Let k be an arbitrary field of characteristic different from 2 containing a square root of −1. In this section, we obtain some information on the 2-Pfister number of quadratic forms of dimension 4 or 6 over k.

The case of anisotropic quadratic forms q ∈ I^{2}(k) of dimension 4 is clear: if q represents 1, then q is a 2-fold Pfister form, so Pf_{2}(q) = 1. On the other hand, if q does not represent 1, then Pf_{2}(q) > 1 and Proposition 2.1 yields Pf_{2}(q) = 2.

We next consider anisotropic forms of dimension 6 in I^{2}(k). Of course, Pf_{2}(q) > 1 for any such form q. If q represents 1, it follows from Proposition 2.1 that Pf_{2}(q) = 2 or 3. The Stiefel-Whitney invariant w_{4}(q) ∈ H^{4}(k, μ_{2}) discriminates between the two cases, as the next proposition shows. (See [8, §3] or [4, §17] for a discussion of Stiefel-Whitney invariants of quadratic forms.)

Proposition 3.1. Let q be an anisotropic quadratic form of dimension 6. Assume q ∈ I^{2}(k) and q represents 1. If w_{4}(q) = 0, then Pf_{2}(q) = 2. If w_{4}(q) ≠ 0, then Pf_{2}(q) = 3.

Proof. In view of Proposition 2.1, it suffices to show that Pf_{2}(q) = 2 if and only if w_{4}(q) = 0. Assume first Pf_{2}(q) = 2 so that

q = ⟨x_{1}, x_{2}, x_{1}x_{2}, y_{1}, y_{2}, y_{1}y_{2}⟩ for some x_{1}, …, y_{4} ∈ k^{×}.

For x ∈ k^{×}, denote by (x) ∈ H^{1}(k, μ_{2}) the cohomology class associated to the square class of x. An explicit computation yields

w_{4}(q) = (x_{1}) ∪ (x_{2}) ∪ (y_{1}) ∪ (y_{2}).

Since q represents 1, the form ⟨1⟩ ⊥ q is isotropic. The 4-fold Pfister form ⟨⟨x_{1}, x_{2}, y_{1}, y_{2}⟩⟩ which contains ⟨1⟩ ⊥ q as a subform is hyperbolic. Therefore, (x_{1}) ∪ (x_{2}) ∪ (y_{1}) ∪ (y_{2}) = 0 by [1, Satz 1.6].

For the converse, let

q = ⟨1, a, b, c, d, abcd⟩ for some a, b, c, d ∈ k^{×}.

Then

w_{4}(q) = (a) ∪ (b) ∪ (c) ∪ (d).

Since w_{4}(q) = 0 by hypothesis, Theorem 1 of [2] shows that the 4-fold Pfister form ⟨⟨a, b, c, d⟩⟩ is hyperbolic. It follows that the 9-dimensional subform q ⊥ ⟨ab, ac, ad⟩ is isotropic, hence q represents a nonzero element of the form a(bx^{2} + cy^{2} + dz^{2}) for some x, y, z ∈ k. Let b′ = bx^{2} + cy^{2} + dz^{2} ∈ k^{×}. Since the form ⟨b, c, d⟩ represents b′, we may find c′, d′ ∈ k^{×} such that

⟨b, c, d⟩ = ⟨b′, c′, d′⟩.

Comparing discriminants, we have bcd ≡ b′c′d′ mod k^{×2}, hence

q = ⟨1, a, b′, c′, d′, ab′c′d′⟩.

The form ⟨1, a, b′⟩ is anisotropic since q is anisotropic, hence the 2-fold Pfister form ⟨⟨a, b′⟩⟩ is anisotropic. On the other hand, the form q ⊥ ⟨ab′⟩ is isotropic since q represents ab′, hence ⟨⟨a, b′⟩⟩ represents a nonzero element of the form c′r^{2} + d′s^{2} + ab′c′d′t^{2}, for some r, s, t ∈ k. Let c′′ = c′r^{2} + d′s^{2} + ab′c′d′t^{2} ∈ k^{×}, and let d′′ ∈ k^{×} be such that

⟨c′, d′, ab′c′d′⟩ = ⟨c′′, d′′, ab′c′′d′′⟩.

Thus

q = ⟨1, a, b′, c′′, d′′, ab′c′′d′′⟩.

Since ⟨⟨a, b′⟩⟩ represents c′′, the 3-fold Pfister form ⟨⟨a, b′, c′′⟩⟩ is hyperbolic, and therefore its 5-dimensional subform ⟨1, a, b′, c′′, ab′c′′⟩ is isotropic. Thus, ⟨1, a, b′, c′′⟩ represents ab′c′′, and we may find u, v ∈ k^{×} such that

⟨1, a, b′, c′′⟩ = ⟨ab′c′′, u, v, uv⟩.

Thus

q = ⟨⟨u, v⟩⟩ + ⟨⟨d′′, ab′c′′⟩⟩,

and Pf_{2}(q) = 2. □

For arbitrary 6-dimensional anisotropic quadratic forms in I^{2}(k), the 2-Pfister number is 2, 3 or 4. Note that scaling has an important effect on the Pfister number although it does not change the Stiefel-Whitney class. Indeed, by Theorem 2.2 and Remark 2.4, if x_{1}, …, x_{5} are independent indeterminates and

q = ⟨x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{1}x_{2}x_{3}x_{4}x_{5}⟩,

then

Pf_{2}(q) = 4 and Pf_{2}(⟨x_{1}⟩q) = 3.

On the other hand,

⟨x_{1}x_{2}x_{3}⟩q = ⟨⟨x_{1}x_{2}, x_{1}x_{3}⟩⟩ + ⟨⟨x_{4}x_{5}, x_{1}x_{2}x_{3}x_{4}⟩⟩,

hence

Pf_{2}(⟨x_{1}x_{2}x_{3}⟩q) = 2.

More generally, the same computation shows that for an arbitrary anisotropic form q ∈ I^{2}(k) of dimension 6, if d ∈ k^{×} is the discriminant of some 3-dimensional subform of q, then Pf_{2}(⟨d⟩q) = 2. In the rest of this section, we give necessary and sufficient conditions on q for Pf_{2}(q) ≤ 3 as well as for Pf_{2}(q) = 2.

As seen before, every quadratic form of dimension 6 in I^{2}(k) is a scalar multiple of a form q with Pf_{2}(q) = 2. Fix a decomposition

q = ⟨⟨a, b⟩⟩ + ⟨⟨c, d⟩⟩ = ⟨a, b, ab, c, d, cd⟩.

To this decomposition is associated the biquaternion algebra D = (a, b)_{k} ⊗ (c, d)_{k}, which is Brauer-equivalent to the Clifford algebra of q, and the orthogonal involution σ on D that is the tensor product of the conjugation involutions on (a, b)_{k} and (c, d)_{k}. The algebra D is division since q is anisotropic, see [6, (16.5)].

Theorem 3.2. For λ ∈ k^{×}, we have Pf_{2}(⟨λ⟩q) = 2 if and only if λ^{2} is the reduced norm of some σ-symmetric element in D, i.e.,

λ^{2} = Nrd_{D}(u) for some u ∈ Sym(D, σ).

Proof. Let (a, b)_{k}^{0} (resp. (c, d)_{k}^{0}) be the k-vector space of pure quaternions in (a, b)_{k} (resp. in (c, d)_{k}). The vector space of σ-skew-symmetric elements in D is

Skew(D, σ) = ((a, b)_{k}^{0} ⊗ 1) ⊕ (1 ⊗ (c, d)_{k}^{0}).

Let p_{σ} be the linear operator on Skew(D, σ) defined by

p_{σ}(x ⊗ 1 + 1 ⊗ y) = x ⊗ 1 − 1 ⊗ y for x ∈ (a, b)_{k}^{0} and y ∈ (c, d)_{k}^{0}.

The formula q_{σ}(s) = s p_{σ}(s) defines a quadratic form on Skew(D, σ), and we have

q_{σ} ≃ ⟨a, b, ab, c, d, cd⟩ = q.

Suppose now Pf_{2}(⟨λ⟩q) = 2. We fix a decomposition

⟨λ⟩q = ⟨⟨a′, b′⟩⟩ + ⟨⟨c′, d′⟩⟩ = ⟨a′, b′, a′b′, c′, d′, c′d′⟩.

The Clifford algebras of q and ⟨λ⟩q are isomorphic, hence we may identify

D = (a′, b′)_{k} ⊗ (c′, d′)_{k}.

Let σ′ be the orthogonal involution on D that is the tensor product of the conjugation involutions on (a′, b′)_{k} and (c′, d′)_{k}. By [6, (2.7)], there is a unit u ∈ Sym(D, σ) such that σ′ = Int(u) ∘ σ, i.e., σ′(x) = uσ(x)u^{−1} for all x ∈ D. On Skew(D, σ′) we may define a linear operator p_{σ′} and a quadratic form q_{σ′} in the same way as p_{σ} and q_{σ} were defined on Skew(D, σ), and we have

q_{σ′} ≃ ⟨λ⟩q.

It is easily seen that Skew(D, σ′) = u Skew(D, σ) = Skew(D, σ)u^{−1}. The linear operator p′ on Skew(D, σ′) defined by

p′(s′) = up_{σ}(s′u) for s′ ∈ Skew(D, σ′)

satisfies

(7)  s′p′(s′) = s′up_{σ}(s′u) = q_{σ}(s′u) ∈ k.

Therefore, by [6, (16.22)], the map p′ is a multiple of p_{σ′}: there exists λ_{1} ∈ k^{×} such that p′ = λ_{1}p_{σ′}. It follows that

(8)  s′p′(s′) = λ_{1}q_{σ′}(s′)  for s′ ∈ Skew(D, σ′),

and (7) shows that the map s′ ↦ s′u is an isometry ⟨λ_{1}⟩q_{σ′} ≃ q_{σ}. Hence ⟨λ_{1}λ⟩q ≃ q and λλ_{1}^{−1} is the multiplier of a similitude of q. By [6, (15.34)], we may find λ_{2} ∈ k^{×} and v ∈ D^{×} such that

(9)  λλ_{1}^{−1} = λ_{2}^{2} Nrd_{D}(v).

On the other hand, for s′ ∈ Skew(D, σ′) we have q_{σ}(s′u)^{2} = Nrd_{D}(s′u) and q_{σ′}(s′)^{2} = Nrd_{D}(s′) by [6, (16.25)], hence (7) and (8) yield

λ_{1}^{2} = Nrd_{D}(u).

Using this equation, we derive from (9):

λ^{2} = λ_{1}^{2}λ_{2}^{4} Nrd_{D}(v)^{2} = Nrd_{D}(λ_{2}vuσ(v)).

Since λ_{2}vuσ(v) ∈ Sym(D, σ), the element λ satisfies the condition in the theorem.

Conversely, assume λ^{2} = Nrd_{D}(u) for some u ∈ Sym(D, σ). Define an orthogonal involution σ′ on D by σ′ = Int(u) ∘ σ. By [6, (7.3)], the discriminant of σ′ is Nrd_{D}(u) = λ^{2}, hence by [6, (15.12)] we may find quaternion subalgebras (a′, b′)_{k}, (c′, d′)_{k} ⊆ D such that

D = (a′, b′)_{k} ⊗ (c′, d′)_{k},

and σ′ is the tensor product of the conjugations on (a′, b′)_{k} and (c′, d′)_{k}. We may then define p_{σ′} and q_{σ′} as above, and we have

(10)  q_{σ′} ≃ ⟨a′, b′, a′b′, c′, d′, c′d′⟩.

On the other hand, define a linear operator p_{0} and a quadratic form q_{0} on Skew(D, σ′) by

p_{0}(s′) = λ^{−1}up_{σ}(s′u), q_{0}(s′) = λ^{−1}q_{σ}(s′u) for s′ ∈ Skew(D, σ′).

By definition, we have

(11)  q_{0} ≃ ⟨λ⟩q_{σ} ≃ ⟨λ⟩q.

Moreover, s′p_{0}(s′) = q_{0}(s′) ∈ k for s′ ∈ Skew(D, σ′), hence p_{0} is a multiple of p_{σ′} by [6, (16.22)]: we have p_{0} = μp_{σ′} for some μ ∈ k^{×}, hence also q_{0} = μq_{σ′}. For s′ ∈ Skew(D, σ′) we have by [6, (16.25)]

p_{0}^{2}(s′) = λ^{−2}up_{σ}(up_{σ}(s′u)u) = λ^{−2} Nrd_{D}(u)p_{σ}^{2}(s′u)u^{−1}.

Since p_{σ}^{2} = Id and Nrd_{D}(u) = λ^{2}, it follows that p_{0}^{2} = Id. Now, we also have p_{σ′}^{2} = Id, hence μ = ±1. Therefore, q_{0} ≃ ⟨±1⟩q_{σ′} ≃ q_{σ′}. By (10) and (11) we have

⟨λ⟩q ≃ ⟨a′, b′, a′b′, c′, d′, c′d′⟩,

hence Pf_{2}(⟨λ⟩q) = 2. □

Note that the group S(q) ⊂ k^{×} of spinor norms of q can be described in terms of D: we have by [6, (15.34)]

S(q) = {λ ∈ k^{×} | λ^{2} ∈ Nrd_{D}(D^{×})}.

Therefore, the following is a direct consequence of Theorem 3.2:

Corollary 3.3. Let q ∈ I^{2}(k) be an anisotropic quadratic form of dimension 6, and let λ ∈ k^{×}. If Pf_{2}(q) = Pf_{2}(⟨λ⟩q) = 2, then λ is a spinor norm of q.

We now turn to a characterization of quadratic forms of dimension 6 with 2-Pfister number at most 3.

Theorem 3.4. Let q ∈ I^{2}(k) be an anisotropic quadratic form of dimension 6. We have Pf_{2}(q) ≤ 3 if and only if there exist a 4-dimensional quadratic form q_{1} over k and scalars μ, μ′, ν ∈ k^{×} satisfying the following conditions:

(i) q ≃ q_{1} ⊥ ⟨μ′⟩⟨⟨ν⟩⟩; (ii) Pf_{2}(q_{1} ⊥ ⟨μ⟩⟨⟨ν⟩⟩) ≤ 2; (iii) ⟨⟨μ, μ′, ν⟩⟩ = 0.

Proof. Suppose Pf_{2}(q) ≤ 3, and let

q = ⟨⟨x_{1}, y_{1}⟩⟩ + ⟨⟨x_{2}, y_{2}⟩⟩ + ⟨⟨x_{3}, y_{3}⟩⟩ = ⟨x_{1}, y_{1}, x_{1}y_{1}, x_{2}, y_{2}, x_{2}y_{2}⟩ + ⟨1, x_{3}, y_{3}, x_{3}y_{3}⟩.

Since the dimension of q is 6, there exists a 2-dimensional form ⟨μ⟩⟨⟨ν⟩⟩ that is a subform of ⟨x_{1}, y_{1}, x_{1}y_{1}, x_{2}, y_{2}, x_{2}y_{2}⟩ and of ⟨1, x_{3}, y_{3}, x_{3}y_{3}⟩. Thus, we can write

(12)  ⟨x_{1}, y_{1}, x_{1}y_{1}, x_{2}, y_{2}, x_{2}y_{2}⟩ = q_{1} ⊥ ⟨μ⟩⟨⟨ν⟩⟩

and

(13)  ⟨1, x_{3}, y_{3}, x_{3}y_{3}⟩ = ⟨μ_{1}, μ_{2}⟩ ⊥ ⟨μ⟩⟨⟨ν⟩⟩

for some 4-dimensional quadratic form q_{1} and some scalars μ_{1}, μ_{2}. Equation (12) readily yields (ii). Comparing discriminants on each side of (13), we see that

⟨μ_{1}, μ_{2}⟩ = ⟨μ′⟩⟨⟨ν⟩⟩ for some μ′ ∈ k^{×}.

Therefore, adding (12) and (13) yields (i). Finally, (13) shows that ⟨μ, μ′⟩⟨⟨ν⟩⟩ represents 1, hence ⟨1, μ, μ′, μν, μ′ν⟩ is isotropic. Since this form is contained in the 3-fold Pfister form ⟨⟨μ, μ′, ν⟩⟩, we have (iii).

Conversely, suppose (i), (ii), and (iii) hold for some 4-dimensional quadratic form q_{1} and some scalars μ, μ′, ν ∈ k^{×}. Since

⟨⟨μ, μ′, ν⟩⟩ = ⟨μ, μν, μ′, μ′ν⟩ ⊥ ⟨1, ν, μμ′, μμ′ν⟩,

condition (iii) yields

⟨μ⟩⟨⟨ν⟩⟩ ⊥ ⟨μ′⟩⟨⟨ν⟩⟩ = ⟨⟨ν, μμ′⟩⟩.

Therefore, we derive from (i) that

q = (q_{1} ⊥ ⟨μ⟩⟨⟨ν⟩⟩) + ⟨⟨ν, μμ′⟩⟩.

Since Pf_{2}(q_{1} ⊥ μ⟨⟨ν⟩⟩) ≤ 2 by (ii), it follows that Pf_{2}(q) ≤ 3. □

References

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[2] J. Kr. Arason, R. Elman and B. Jacob, Fields of cohomological 2-dimension three, Math. Ann. 274 (1986), no. 4, 649–657.

[3] P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension, spinor groups, and quadratic forms, to appear in Ann. Math.

[4] S. Garibaldi, A. Merkurjev and J.-P. Serre, Cohomological invariants in Galois cohomology, Amer. Math. Soc., Providence, RI, 2003.

[5] B. Kahn, On “horizontal” invariants attached to quadratic forms, pp. 21–33, Hindustan Book Agency, Delhi, 2005.

[6] M.-A. Knus et al., The book of involutions, Amer. Math. Soc., Providence, RI, 1998.

[7] T. Y. Lam, Introduction to quadratic forms over fields, Amer. Math. Soc., Providence, RI, 2005.

[8] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318–344.

[9] R. Parimala and V. Suresh, On the length of a quadratic form, in Algebra and number theory, pp. 147–157, Hindustan Book Agency, Delhi, 2005.

Department of Mathematics and Computer Science, Emory University, 400 Dowman Drive, Atlanta, Georgia 30322 USA

    E-mail address: parimala@mathcs.emory.edu

Department of Mathematics and Statistics, University of Hyderabad, Gachibowli, Hyderabad - 500046 India

    E-mail address: vssm@uohyd.ernet.in

Département de mathématique, Université catholique de Louvain, chemin du cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium

    E-mail address: jean-pierre.tignol@uclouvain.be