Splitting Patterns of Quadratic Forms

Jürgen Hurrelbrink* and Ulf Rehmann

Published in: Mathematische Nachrichten, 176 (1995), 111–127.

Introduction.

Let q be an anisotropic quadratic form of dimension n ≥ 2 over a field k of characteristic ≠ 2. The splitting pattern of q is made up from all distinct Witt indices of q which occur over field extensions of k. The entries of the splitting pattern of q are called the higher Witt indices of q.

For excellent quadratic forms q, the splitting pattern depends only on the dimension n of q, and is known explicitly for every n. This goes back to the originating work of M. Knebusch [10, 11] and has been studied further in [5]. However, surprisingly little is known about the splitting pattern of quadratic forms q, in general, as soon as there are at least 2 non-zero Witt indices in the splitting pattern of q.

In section 1 we investigate general constraints on splitting patterns. In theorem 1.6 we describe in which way the splitting pattern of a quadratic form is influenced by its Clifford algebra. For this we use the index reduction formulas of A. S. Merkurjev [14] as the crucial tool. It turns out that the splitting of the Clifford algebra determines the “high end” of the splitting pattern.

On its “low end”, we find that the first higher Witt index i_{1}(q) of q always is less than or equal to the first higher Witt index of the anisotropic excellent forms of the same dimension as q. In particular, i_{1}(q) is less than or equal to the excess of dim q over the biggest 2-power which is strictly smaller than dim q. If equality holds, then q is a “stable Pfister neighbor”; that is, there is a field extension K of k such that q_{K} is an anisotropic Pfister neighbor.

Similar results on i_{1}(q) have earlier been obtained by D. W. Hoffmann [4].

Here we give a short proof based on Theorem 1.7, which describes a useful criterion to determine when an anisotropic form q is an orthogonal summand of an anisotropic Pfister form π. Corollary 1.8 establishes the existence of a unique field extension K of k in a generic splitting tower of π ⊥ −q such that π_{K} is anisotropic and has the anisotropic kernel (q_{K})_{an} of q as an orthogonal summand.

In section 2 we study the “algebraic” splitting pattern; that is, we look at the possible Witt indices which can occur over algebraic extensions of k. In general, the algebraic splitting pattern is different from the splitting pattern. However, we show that for given dimension n there is a number t such that, for every form q of dimension n over any field k, the splitting pattern of q is equal to the algebraic splitting pattern of q_{K}, where K is a purely transcendental field extension of k of degree t.

In section 3 we approach the wide open case of quadratic forms of height 2; that is, with exactly 2 non zero Witt indices in the splitting pattern of q; see, in particular, theorems 3.4 and 3.11 for so-called good forms of degree 3 and forms of degree 2 that are not good, respectively. The general theorem 3.9 makes explicit the splitting pattern of good forms of arbitrary degree, assuming that the conjectures by B. Jacob and M. Rost [6] on higher cohomological invariants are true. Under this assumption those forms have always twice the dimension of their leading form, and over a suitable field extension they are (still anisotropic) a product of a four dimensional form and a Pfister form. This also answers a question of M. Knebusch [11, 10.6, p. 30].

* partially supported by a DAAD grant, Study Visit at the Sonderforschungsbereich 343, University of Bielefeld, summer 1993.

We gladly acknowledge most helpful conversations with D. W. Hoffmann and J. Morales on section 1 and B. Kahn on section 3. We also thank I. Kersten very much for several improvements and corrections.

0. Setup.

Let q be a regular quadratic form of index i(q) and let

i_{0} := i(q) < i_{1} < … < i_{h−1} < i_{h} = [n/2]

denote all distinct Witt indices which occur for the quadratic forms q_{L} := q⊗L with L running through all field extensions of k.

The natural number h is called the height of q. The (h + 1)-tuple (i_{0}, i_{1}, …, i_{h}) of strictly increasing integers is called the splitting pattern of q (cf. [5, 1.1]). If necessary, we will write i_{ν}(q) and h(q) instead of i_{ν} and h. We now illustrate the extreme cases h = 1 and h = m := [n/2].

0.1 Example (Height 1; cf. [10, Thm. 5.8, p. 81]). An anisotropic quadratic form q of dimension ≥ 2 over k has splitting pattern (0, m) for some m ∈ ℕ if and only if q is similar to an orthogonal summand of codimension ≤ 1 of a Pfister form over k.

0.2 Example (Height m; cf. [10, Ex. 5.7, p. 80]). Let q = ⟨X_{1}, X_{2}, …, X_{n}⟩ with indeterminates X_{i} over k = F (X_{1}, …, X_{n}) for any field F. Then q has splitting pattern

(0, 1, 2, …, m − 2, m − 1, m).

Given m ∈ ℕ, there are 2^{m−1} different tuples (0, …, m) of strictly increasing integers. What about realizing them as splitting patterns? By example 0.1, there is no quadratic form q with splitting pattern (0, 5) or (0, 6), say, since there do not exist Pfister forms of appropriate dimensions.

The basic question of our paper is: What are the splitting patterns of quadratic forms?

The above observation concerning splitting patterns of quadratic forms of height 1 yields immediately that a tuple (0, …, m − a, m) can be realized to be a splitting pattern of some quadratic form of arbitrary height only if a or a − 1 is a 2-power.

We will look for further restrictions on splitting patterns and will comment on the next-to-extreme cases h = 2 and h = m − 1.

By the disciminant d(q) of any quadratic form q we always mean its signed discriminant. The Witt invariant is defined as in [13, Ch. V.3, p. 120]. We recall the standard notation for Pfister forms: For n > 0 and a_{1}, …, a_{n} ∈ k^{∗} the 2^{n}-dimensional quadratic form ⟨⟨a_{1}, …, a_{n}⟩⟩ := ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{n}⟩ is called an n-fold Pfister form. Sometimes it makes sense to call the form ⟨1⟩ a 0-fold Pfister form.

For the definition of the leading field and the leading form of a quadratic form we refer to [10, 5.4, p. 79 and 5.9, p. 82]. The degree of a form q is defined in [10, §6, p. 88].

1. General constraints on splitting patterns We will present an analog to example 0.2. Consider the rational function field k = F (X_{1}, …, X_{n−1}) with indeterminates X_{1}, … X_{n−1} for any field F of characteristic ≠ 2, let c ∈ F^{∗}.

1.1 Lemma. The n-dimensional quadratic form

q = ⟨X_{1}, X_{2}, …, X_{n−1}, cX_{1}X_{2} ⋯ X_{n−1}⟩

is anisotropic over k for any n ≥ 3.

Proof. Assume that q is isotropic over k. Then there are relatively prime polynomials p_{1}, …, p_{n} in F [X_{1}, … X_{n−1}] with

(∗)  X_{1}p_{1}^{2} + X_{2}p_{2}^{2} + ⋯ + X_{n−1}p_{n−1}^{2} = −cX_{1}X_{2} ⋯ X_{n−1}p_{n}^{2}.

As the form ⟨X_{i_{1}}⟩ ⊥ … ⊥ ⟨X_{i_{r}}⟩ is anisotropic over k for every subset {i_{1}, …, i_{r}} of {1, …, n − 1} we conclude, using the substitution X_{i} = 0, that X_{i} divides p_{j} for every j ∈ {1, …, n − 1}, j ≠ i. Thus X_{j}p_{j}^{2} = X_{j}(∏_{j≠i} X_{i}^{2})p_{j}^{2} for some p̃_{j} ∈ F [X_{1}, …, X_{n−1}] for all j = 1, …, n − 1. So every summand in (∗) is divisible by X := X_{1} ⋯ X_{n−1}. Dividing (∗) by X yields

(∗∗)  X̂_{1}p_{1}^{2} + … + X̂_{n−1}p_{n−1}^{2} = −cp_{n}^{2}

where X̂_{i} := X/X_{i} = Π_{j≠i}X_{j}. From (∗∗) it follows that X_{n−1} divides p_{n} if and only if it divides p̃_{n−1}, but this cannot happen since the p_{i} are assumed to be relatively prime. By substituting X_{n−1} = 0 we obtain an equation X̂_{n−1} ˜p̃_{n−1}^{2} = −cp_{n}^{2} with non-zero ˜p̃_{n−1}, p̃_{n} ∈ F [X_{1}, …, X_{n−2}]. Now this implies that −X̂_{n−1}/c is a square in k. But for n ≥ 3 this contradicts the fact that F [X_{1}, …, X_{n−1}] is a unique factorization domain.

Let us apply this lemma to the case n = 2m and c = (−1)^{m} to obtain:

1.2 Proposition. Let k = F (X_{1}, …, X_{2m−1}) with indeterminates X_{i} over F and 2 ≤ m. Then q = ⟨X_{1}, X_{2}, …, X_{2m−1}, (−1)^{m}X_{1}X_{2} ⋯ X_{2m−1}⟩ has height m − 1 and splitting pattern (0, 1, 2, …, m − 2, m). Moreover, the Clifford algebra C(q) of q is isomorphic to M_{2}(D), with a central division algebra D that is a tensor product of m − 1 quaternion algebras.

Proof. The 2m-dmional quadratic form q is anisotropic over k by Lemma 1.1. We have arranged for q to have determinant (−1)^{m}, which means that its discriminant is 1. Thus if, for some extension L of k, the form q_{L} splits off at least m − 1 hyperbolic planes, then q_{L} splits completely. That is, m − 1 does not appear as a Witt index over any extension of k.

Hence we are done if m = 2, obtaining splitting pattern (0, 2). We proceed by induction. We write q_{m} := q for m ≥ 3 and assume that the proposition has been proved for all q_{m′} with m′ < m. Then over L = k(\sqrt{−X_{2m−2}X_{2m−1}}) the form q_{m} is isometric to q_{m−1} ⊥ ⟨X_{2m−2}, −X_{2m−2}⟩. Thus it splits off exactly one hyperbolic plane, since by Lemma 1.1 the first summand is still anisotropic over L. Using the induction hypothesis we get our first claim.

To prove our statement about C(q) we write a_{m} = (−1)^{m}X_{1}X_{2} ⋯ X_{2m−1}, Y = X_{2m−2}/a_{m−1}, Z = X_{2m−1}/a_{m−1} and observe that

q_{m−1} ⊥ (−a_{m−1})⟨⟨−Y, −Z⟩⟩ ≅ q_{m} ⊥ ⟨1, −1⟩,

which yields C(q_{m−1}) ⊗ M_{2}(Q) ≅ M_{2}(C(q_{m})) for some quaternion skew field Q over k [13, Ch. V, 2.7, p. 113 and 3.3, p. 116]. Clearly the index of C(q_{m}) is twice the index of C(q_{m−1}), and the assertion follows by induction on m.

Contrasting Proposition 1.2, the reader notices that no odd dimensional form has splitting pattern (0, 1, 2, …, m − 2, m) since no 5-dimensional form has height 1.

More generally, we can see that, for a ≥ 1, no odd dimensional form q can have a splitting pattern (0, 1, 2, …, m − 2^{a}, m) since this would require the existence of an anisotropic form of dimension 2^{a+1} + 1 of height 1 contradicting 0.1.

An even dimensional form with splitting pattern (0, 1, 2, …, m − 2^{a}, m) and a ≥ 2 would require the existence of an anisotropic form q of discriminant 1, Witt invariant 1 (by 1.6 below), height 2 and splitting pattern (0, 1, 2^{a} +1), hence of dimension 2^{a+1} +2. This seems us to be unlikely, especially because of our results in §3. We can exclude the case a = 2 since then dim q = 10, in which case it follows from [15, Proof of Satz 14, No. 5, p. 123] that a form q having invariants as mentioned above is necessarily isotropic.

1.3 Question. Given a ≥ 3, is there an even dimensional form q over some field k with splitting pattern (0, 1, 2^{a} +1)? If the answer is no as we expect, is there a general lower bound on i_{1}(q) (independent of the field k) for even dimensional anisotropic forms q of height 2 as an increasing function of the degree? (3.9 below yields this under certain assumptions for excellent resp. “good” forms of height 2.)

Next we will exhibit constraints on the first non-zero entry of the splitting pattern of a quadratic form.

We begin with excellent forms as introduced by Knebusch [11, 7.7, p. 3]. For those forms, the splitting patterns depend only on the dimension of the form. Let i_{1}(n) and h(n) stand for the first non-zero Witt index and the height of all n-dimensional (anisotropic) excellent forms of dimension n over any field k (of characteristic ≠ 2).

Given j ∈ ℕ, what is the best possible lower bound for i_{1}(n), where n ranges over all dimensions with h(n) = j? An answer is given by the following proposition which sharpens 3.1 in [5].

1.4 Proposition. Let h ≥ 1. Then the natural number

n_{h} := \{ \begin{matrix}(2^{h+2} − 2)/3 & if h is odd, \\ (2^{h+2} − 1)/3 & if h is even \end{matrix}

is the minimal dimension of an excellent form of height h. Moreover we have i_{1}(n) ≥ n_{h(n)−2} for any n ∈ ℕ with h(n) ≥ 3, with equality for excellent forms of dimension n = n_{h}.

Proof. We prove the first statement by induction on h. Obviously n_{1} = 2, and 5 is the lowest dimension of an excellent form of height 2, hence n_{2} = 5 and we are done for h ≤ 2. We now assume that h ≥ 3, and that the first statement has been shown for excellent forms of height < h. We write h = 2u + ε with ε = 0 or ε = 1 and find

(1)  n_{h} = 2^{ε}(4^{u+1} − 1)/3 = 2^{ε}(∑_{i=0}^{u} 4^{i}) = 2^{ε}4^{u} + n_{h−2} = 2^{h} + n_{h−2}.

If q is an excellent form of height h ≥ 2 and if q_{2} denotes its second complementary form it follows from [5, 2.6 and 2.10] that

(2)  dim q = 2^{a} − 2^{b} + dim q_{2}

with a > b ≥ h. Clearly q_{2} has height h − 2. Hence, using the induction hypothesis and (1), we obtain from (2) that dim q ≥ 2^{b+1} − 2^{b} + dim q_{2} ≥ 2^{h} + n_{h−2} = n_{h}. Thus the first statement is proved.

It follows from [5, 1.8] that the dimension of the first complementary form q_{1} of q is given – in terms of (2) – by 2^{b} − dim q_{2}, and hence we obtain

(3)  i_{1}(n) = (dim q − dim q_{1})/2 = (2^{a} − 2^{b} + dim q_{2} − 2^{b} + dim q_{2})/2
= 2^{a−1} − 2^{b} + dim q_{2} ≥ dim q_{2} ≥ n_{h(n)−2}.

If n = n_{h}, then a comparison of (1) and (2) together with the equality dim q_{2} = n_{h−2} shows that 2^{a} − 2^{b} = 2^{h}, hence (3) holds with an equality everywhere.

1.5 Remark. In particular, we have i_{1}(n) > 1 for all n with h(n) ≥ 3.

We now study the influence of the splitting behavior of the Clifford algebra C(q) on the splitting pattern of an arbitrary regular form q.

Let us remark that the function field k(q) of a quadratic form q is a purely transcendental extension followed by a quadratic extension, hence if a central simple algebra over k is tensored by k(q), then its index goes down by a factor of at most 2.

It follows from [13, Ch. V, Thm. 2.5, p. 111] that in case dim q > 0 even and d(q) = 1 the Clifford algebra of q is of the form C(q) ≅ M_{2}(E(q)) for some central simple k-algebra E(q). In the other cases we define E(q) as follows. If dim q is even and d(q) ≠ 1, then E(q) := C(q). If dim q is odd, then E(q) := C_{0}(q), the even part of the Clifford algebra.

If q ≅ ⟨1, −1⟩ ⊥ q′ and q′ ≠ 0, then E(q) ≅ M_{2}(k) ⊗ E(q′) ≅ M_{2}(E(q′)) by [13, Ch. V, 2.7, 2.9, p. 113f].

Hence if E(q) is a skew field ≠ k, then q is anisotropic.

1.6 Theorem. Let q be anisotropic and let E(q) ≅ M_{2^{r}}(D) for some skew field D ≅ k̸ and some r ≥ 0.

i) Let r > 0. Then D_{k(q)} is a skew field, i_{1}(q) ≤ r, and there is some ν such that i_{ν}(q) = r. ii) Let r = 0. Then E(q) is a skew field, and this also holds for all higher anisotropic kernels of q. If dim q is odd or d(q) ≠ 1, then the splitting pattern of q is (0, 1, 2, …, m− 1, m) and E(q) is a product of m quaternion skew fields. If dim q even and d(q) = 1, then the splitting pattern of q is (0, 1, 2, …, m − 2, m) and E(q) is a product of m − 1 quaternion skew fields. (Here m = [dim q/2].)

Proof. For i = i_{1}(q) and q′ = (q_{k(q)})_{an} we have q_{k(q)} ≅ ⟨1, −1⟩^{i} ⊥ q′, hence by [13, Ch. V, 2.7, 2.9, p. 113f]

(∗) M_{2^{r}}(D_{k(q)}) ≅ E(q_{k(q)}) ≅ M_{2^{i}}(k(q)) ⊗_{k(q)} E(q′) ≅ M_{2^{i}}(E(q′)) for q′ ≠ 0.

i) If r > 0, then it follows from Merkurjev’s index reduction theorems [14, Thm. 1, 2, 3, p. 218] that D_{k(q)} is a skew field. Therefore i_{1}(q) = i ≤ r by (∗). Repeating this procedure yields our assertion.

ii) If r = 0, then it follows from (∗) that D_{k(q)} is not a skew field, but then its index is exactly half of the index of D ≅ E(q). Hence i must be 1 unless d(q) = 1 and dim q = 4, in which case i = 2. An induction proves the result.

1.7 Theorem. Let π be an anisotropic Pfister form, let q be any regular form with dim q < dim π, and let q̃:= π ⊥ −q. Then i(q̃) + i(q) ≤ dim q, and the following are equivalent.

i) i(q̃) + i(q) = dim q. ii) π_{k(_{q_{an}})} is isotropic. iii) q_{an} is an orthogonal summand of π.

Proof. We have π ⊥ (−q)_{an} ≅ q̃_{an} ⊥ ℍ^{δ} with δ := i(q̃) − i(q). Since π is anisotropic this yields δ ≤ dim q_{an}, which is equivalent to i(q̃) + i(q) ≤ dim q. There are regular subforms π′ ⊆ π and q′ ⊆ q_{an} each of codimension δ such that π′ ⊥ (−q′) = q̃_{an}.

If i) holds, then δ = dim q_{an}, hence q′ = 0 and q̃_{an} must be an orthogonal summand of π which implies ii).

We show that ii) implies iii). Since dim q < dim π we have π′ ≠ 0, hence there is some s ∈ k^{∗} represented by π and by q̃_{an}. Let k′ := k(q̃_{an}). If π_{k′} is isotropic, then it is hyperbolic. By [12, 7.4, p. 22], the form q̃_{an} = s^{2}q̃_{an} is an orthogonal summand of π, hence there is a regular form q_{1} such that π = q̃_{an} ⊥ q_{1}. By Witt cancellation we get q_{1} ⊥ (−q)_{an} = ℍ^{δ} and thus δ = dim q_{an}. This implies q′ = 0 and therefore q̃_{an} = π′ ⊆ π, hence the orthogonal complement of π′ in π is isomorphic to q_{an}.

If iii) holds, then δ = dim q_{an} which yields i).

The notion of a generic splitting tower of a quadratic form has been introduced by Knebusch in [10, p. 78]. However, we here will use the slightly more general version of [5, 1.1, p. 184].

1.8 Corollary. In any generic splitting tower of q̃ there is a (unique) field extension K of k such that π_{K} is anisotropic and (q_{K})_{an} is an orthogonal summand of π_{K}.

Any two such field extensions are k-equivalent, i. e. they are k-specializations of each other.

Proof. A generic splitting tower {K_{ν}} of q̃ can be constructed inductively by K_{0} := k, K_{ν+1} := K_{ν}((q̃_{K_{ν}})_{an}) (cf. [10, p. 78]).

Take ν maximal such that π_{K_{ν}} is anisotropic. Then π_{K_{ν+1}} is isotropic and we can apply 1.7 with k replaced by K := K_{ν} to obtain that (q_{K})_{an} is an orthogonal summand of π_{K}.

If now K and K′ are two fields from generic splitting towers of q̃ such that π_{K} and π′_{K} are anisotropic and (q_{K})_{an} (resp. (q_{K′})_{an}) is an orthogonal summand of π_{K} (resp. π_{K′}), then, by 1.7, we obtain i(q̃_{K}) + i(q_{K}) = dim q = i(q̃_{K′}) + i(q_{K′}). We may assume that i(q̃_{K}) ≥ i(q̃_{K′}), hence there is a k-specialization K′ → K ∪ {∞} which implies i(q_{K}) ≥ i(q_{K′}). Therefore we get i(q̃_{K}) = dim q − i(q_{K}) ≤ dim q − i(q_{K′}) = i(q̃_{K′}) and hence i(q̃_{K}) = i(q̃_{K′}), obtaining a k-specialization K → K′ ∪ {∞}.

1.9 Corollary. Let dim π = 2^{r+1} and K be as in Corollary 1.8.

i) If dim q − i(q_{K}) ≤ 2^{r}, then K(π_{K}) is k-equivalent to k(π). ii) Assume q_{k(π)} is anisotropic. If either dim q ≤ 2^{r} or dim q = 2^{r} + i_{1}(q), then q_{K} is an orthogonal summand of the anisotropic Pfister form π_{K}.

Proof. By 1.7, 1.8 we have dim q − i(q_{K}) = i(q̃_{K}). Let V denote the variety of (min {2^{r}, i(q̃_{K})})-dimensional totally isotropic subspaces of q̃. Then V has a rational point over k(π), since i(q̃_{k(π)}) ≥ 2^{r}. Hence it follows from [9, Thm. 3.10, p. 46, and 5.2, p. 58] that k(π)(V_{k(π)}) ≅ k(V)(π_{k(V)}) is purely transcendental over k(π), and that there is a k-specialization k(V) → K ∪ {∞}. Hence i(q̃_{k(V)}) ≤ i(q̃_{K}) with equality if and only if k(V) and K are k-equivalent.

i) We have i(q̃_{k(V)}) = i(q̃_{K}), hence the fields K(π_{K}) and k(V)(π_{k(V)}) are kequivalent. Thus i) follows since k(V)(π_{k(V)}) is purely transcendental over k(π).

ii) By 1.8, we have to show that i(q_{K}) = 0. If K and k(V) are k-equivalent, then i(q_{K}) = i(q_{k(V)}) = 0, since q_{k(π)} is anisotropic and k(V)(π_{k(V)}) is purely transcendental over k(π). Otherwise 2^{r} ≤ i(q̃_{k(V)}) < i(q̃_{K}) ≤ i(q̃_{K}) + i(q_{K}) = dim q = 2^{r} + i_{1}(q) which implies i(q_{K}) ≤ i_{1}(q) − 1, hence i(q_{K}) = 0.

1.10 Lemma. Let k′ := k(X_{0}, …, X_{r}) and π := ⟨⟨X_{0}, …, X_{r}⟩⟩. Then π is anisotropic, and k′(π) is unirational over k. If q is a regular form over k, then q and q_{k′(π)} have the same splitting pattern.

Proof. Since π is isotropic over k′′ := k′(⟨1, X_{0}⟩) we find that k′′(π) is purely transcendental over k′′. As k′′ is purely transcendental over k, it follows that k′(π) ⊂ k′′(π) is unirational over k.

If L is a field extension of k, then the free composite L′ := Lk′ is isomorphic to the rational function field L(X_{0}, …, X_{r}) and, by the above, L_{1} := L′(π_{L′}) is unirational over L and contains k′(π). Hence i(q_{L}) = i(q_{L_{1}}). This shows that every Witt index which occurs over an extension of k also occurs over an extension of k′(π).

Except of the statements on splitting patterns, the following three corollaries 1.11, 1.12, 1.13 are contained in [4, Thm. 2, Cor. 3 and Thm. 1]. We here give shorter proofs based on Thm. 1.7.

1.11 Corollary. Let q be an anisotropic form with either dim q ≤ 2^{r} or dim q = 2^{r} + i_{1}(q) < 2^{r+1}. Then there is a field extension K of k and an anisotropic (r + 1)- fold Pfister form π over K such that q_{K} ⊆ π. If dim q ≤ 2^{r}, then K and π can be chosen such that, for every regular form q′ over k, the forms q′ and q′_{K(π)} have the same splitting pattern.

Proof. We apply Corollary 1.9, with k replaced by k′ from Lemma 1.10 and with the particular π defined there and with q replaced by q_{k′}. The unirationality of the field k′(π) over k as stated in Lemma 1.10 gives the anisotropy of q_{k′(π)} required by 1.9 ii).

1.12 Corollary (Hoffmann, cf. [4, Thm. 1]). Let q, q′ be regular forms such that q′ is anisotropic and, for some natural number r, we have dim q′ ≤ 2^{r} < dim q. Then q′_{k(q)} is anisotropic. More generally, q′ and q′_{k(q)} have the same splitting pattern.

Proof. If q is isotropic, then k(q) is a purely transcendental field extension of k [10, 3.9, p. 72] and there is nothing to show. Otherwise take K, π as in Corollary 1.11, but for the form q′. If q′_{k(q)} is isotropic, then π_{K(q_{K})} splits, hence q_{K} is a subform of π by [10, 4.5, p. 75]. As q_{K(π)} is anisotropic this implies dim q ≤ 2^{r}.

Now if L is any field extension of k, then L(q_{L}) is a field extension of k(q), and (q′_{L})_{an} stays anisotropic over L(q_{L}) by what we just showed applied to (q′_{L})_{an} and q_{L}. Hence the splitting patterns of q′ and q′_{k(q)} are the same.

1.13 Corollary. Let q be an anisotropic quadratic form of dimension n. Then we have i_{1}(q) ≤ i_{1}(n). If equality holds, then there is a field extension K of k such that q_{K} is an anisotropic Pfister neighbor. Conversely, if q is a Pfister neighbor with complementary form q′, then −q′_{k(q)} is the anisotropic kernel of q_{k(q)} and i_{1}(q) = i_{1}(n) holds.

Proof. By [5, 2.8, p. 187], i_{1}(n) is the difference of n and the greatest 2-power which is strictly smaller than n. Hence the first part of the theorem follows, since, by 1.12, no subform of q of codimension i_{1}(n) becomes isotropic over k(q). If equality holds, we apply 1.11 if dim q is not a power of 2, otherwise we apply 0.1.

If q is a Pfister neighbor with complementary form q′ we set d = dim q − 2^{r}. Then dim q′ = 2^{r} −d, hence q′_{k(q)} is anisotropic by 1.12. It then follows from [5, 2.9 (i), p. 187] that −q′_{k(q)} is the anisotropic kernel of q_{k(q)}. But this means that i_{1}(q) = d = i_{1}(n).

1.14 Remark. Let q be anisotropic of dimension n. If q_{K} becomes an anisotropic Pfister neighbor for some field extension K of k, then it is not necessarily true that i_{1}(q) = i_{1}(n) as can be seen by some combination of 1.18 and, say, 0.2. In fact 1.18 implies that there are fields k such that every anisotropic quadratic form becomes an anisotropic Pfister neighbor over some field extension of k. As B. Kahn has mentioned to us, i_{1}(q) = i_{1}(n) if and only if there exists a field extension K of k such that q_{K} is an anisotropic Pfister neighbor and h(q_{K}) = h(q).

The case l = 1 of the following statement has also been observed by D. W. Hoffmann.

1.15 Corollary. Let q be anisotropic and dim q < 2^{r}. Then, for any natural number l, there is a field extension K of k and an anisotropic form q̃ over K with ñ := dim q̃ < 2^{r+l} which is a q_{K}-extension of order l in the sense of [5, 2.2, p. 185]; that is, there is a sequence q̃ = q_{0}, q_{1}, …, q_{l} = q_{K} of anisotropic forms over K such that q_{j} is a Pfister neighbor with complementary form q_{j+1} for all j < l. If the splitting pattern of q is (0, i_{1}, …, i_{h}), then the splitting pattern of q̃ is given by (0, i_{1}(ñ), …, i_{l}(ñ), i_{1} + i_{l}(ñ), …, i_{h} + i_{l}(ñ)). Every (generic) splitting field of q_{K} is a (generic) splitting field of q̃.

Proof. Assume l = 1. Corollary 1.11 implies the existence of a field extension K of k and an r + 1-fold anisotropic Pfister form π over K such that q_{K} is a subform of π, hence it is the complementary form of some Pfister neighbor q̃ of π. Since dim q < 2^{r} it follows from 1.11 that K can be choosen such that q and q_{K} have the same splitting pattern. Clearly dim q̃ > 2^{r}, hence q and q_{K(_{q})} have the same splitting pattern by 1.12. Together with the second half of 1.13 this now yields the result on the splitting pattern. The statement about the (generic) splitting field follows from [5, 2.4, 2.5, p. 186]. A straightforward induction now proves the case of arbitrary l.

1.16 Example. The splitting patterns of anisotropic forms q with dim q = n ≤ 9 are given by

n = 2  (0, 1)
n = 3  (0, 1)
n = 4  (0, 2), (0, 1, 2)

n = 5  (0, 1, 2)
n = 6  (0, 2, 3), (0, 1, 3), (0, 1, 2, 3)
n = 7  (0, 3), (0, 1, 2, 3)
n = 8  (0, 4), (0, 2, 4), (0, 1, 2, 4), (0, 1, 3, 4), (0, 1, 2, 3, 4)

and possibly (0, 2, 3, 4)

n = 9  (0, 1, 4), (0, 1, 2, 3, 4).

Explanation. The first entry in each line above is the excellent splitting pattern, as can easily be checked from [11, 7.10, p. 4] or [5, 2.8, p. 187].

Everything is clear for dimension ≤ 3. For n = 4 both patterns occur by examples 0.1 and 0.2. For n = 5, (0, 2) fails to be a splitting pattern by 0.1. For n = 6, the pattern (0, 1, 3) occurs by 1.2, (0, 2, 3) is the excellent pattern, (0, 1, 2, 3) is realizable by 0.2, but (0, 3) is not by 0.1.

For n = 7, (0, 3) and (0, 1, 2, 3) occur by 0.1 and 0.2, respectively. The tuple (0, 1, 3) fails to be a splitting pattern since (0, 2) does not occur for n = 5. We show that the only remaining tuple (0, 2, 3) also is not realizable. A corresponding form q cannot be excellent, hence its Witt invariant c(q) is non trivial by [11, p. 11]. By 1.6 we conclude that c(q) is given by a single quaternion skew field, hence the leading form of q is defined over k by [11, 9.8, p. 23]. But this contradicts the initial remark of [11, §10, p. 27].

For n = 8, (0, 4) occurs by 0.1; for (0, 2, 4) we refer to Remark 3.3 below, and for (0, 1, 2, 4) to Prop. 1.2. Moreover, the tuple (0, 1, 4) is not realizable since (0, 3) is not realizable in dimension 6. It is left to determine the splitting patterns of forms of degree 1; that is, of discriminant ≠ 1. We can exclude (0, 3, 4) as a pattern by Remark 3.2. The tuple (0, 1, 3, 4) is realizable by the form ⟨1, 1, 1, 1, 1, 1, 1, X⟩ over ℚ(X). For the tuple (0, 1, 2, 3, 4) we can refer to 0.2 or 1.6 ii).

For n = 9, (0, 1, 4) is the excellent pattern, (0, 1, 2, 3, 4) occurs by 0.2. All other six tuples can be excluded, namely: (0, 4), (0, 2, 4), (0, 3, 4), and (0, 2, 3, 4) are not realizable as splitting patterns since i_{1}(9) = 1 by Corollary 1.13. Finally, (0, 1, 2, 4) does not occur since (0, 2) is not realizable in dimension 5, and (0, 1, 3, 4) does not occur since (0, 2, 3) is not realizable in dimension 7.

1.17 Definition. A regular anisotropic form q over k is called stably excellent if there is a field extension K of k such that q_{K} is anisotropic and excellent.

1.18 Proposition.

i) If k is a formally real field and if q = ⟨a_{1}, …, a_{n}⟩ where all the elements a_{i} are positive under some real embedding of k, then q is stably excellent. ii) An anisotropic form q is stably excellent if dim q ≤ 5 or if k is a number field.

Proof. i) Let K = k(\sqrt{a_{1}}, …, \sqrt{a_{n}}). Then q_{K} ≃ n × ⟨1⟩; hence q_{K} is anisotropic and excellent.

ii) For dim q ≤ 3 there is nothing to show since every such form is excellent. A form q of dimension 4 and of discriminant d ≠ 1 is anisotropic if and only if q_{k(\sqrt{d})} is anisotropic, but the latter has discriminant 1 and hence is similar to the norm form of a quaternion skew field, hence to a Pfister form and therefore it is excellent. If dim q = 5, then 1.13 yields i_{1}(q) = i_{1}(n) = 1, and there is a field extension K of k such that q_{K} is an anisotropic Pfister neighbor. Then the first kernel form of q_{K} is of dimension 3 and hence excellent, therefore q_{K} is excellent. If k is a number field and dim q ≥ 5 then, by the theorem of Minkowski-Hasse, the anisotropy of q implies that there is a real embedding of k over which q is (positive or negative) definite and hence anisotropic. Now the assertion follows from i).

1.19 Remark. The splitting pattern of an excellent 6-dimensional form is (0, 2, 3). Hence no 6-dimensional form with pattern (0, 1, 3) is stably excellent; compare Theorem 3.11 and Remark 3.13 below.

2. Algebraic splitting patterns

2.1 Definition. The algebraic splitting pattern of a regular quadratic form q over k is the subsequence of its splitting pattern whose entries can be obtained as the Witt indices of q_{K} for algebraic field extensions K of k.

2.2 Example. Let q denote the sum of 9 squares over ℚ, which is clearly anisotropic, and let F be any number field. An application of the Minkowski-Hasse Theorem shows that if q_{F} is isotropic, then F has level ≤ 4 and q splits totally; that is, q has Witt index 4. But the transcendental extension k(q) is a field extension of level 8 and hence the Witt index of q_{k(q)} is just 1.

This yields that the algebraic splitting pattern of q is (0, 4), whereas the splitting pattern is (0, 1, 4). A complete determination of the algebraic splitting pattern of excellent forms over ℚ is given in [5, Thm. 3.5].

Hence the algebraic splitting pattern in general differs from the splitting pattern. However, there is always a purely transcendental field extension K of k (so that the Witt indices of q_{K} and q are the same) such that the algebraic splitting pattern of q_{K} is the same as the splitting pattern of q, as we will see below. The transcendence degree of K over k can be chosen in a fairly uniform manner depending just on dim q and not on the underlying field k or on the form q itself.

More precisely we have the following.

2.3 Theorem. For any natural number n, there is a natural number t = t(n) with the following property:

Let k be any field and let q be any regular quadratic form over k of dimension n. Then if K is a purely transcendental extension of k of degree t, every Witt index which does occur for q over some extension of k is also obtained as the Witt index of q_{K′} for a finite separable extension K′ of K.

A possible choice for t is the dimension of the unipotent radical of a Borel subgroup of SO_{q} × k_{s}, where SO_{q} is the special orthogonal group of q, and k_{s} is some separable algebraic closure of k.

Proof. The number t as described in the last part of the theorem is exactly half of the number of the roots in a root system of G := SO_{q} × k_{s} and depends on the dimension of q only, not on the choice of the field k or of q itself.

It follows from [9, 5.2 and 5.3] that a generic splitting tower {K_{i}}_{i=0,...,h} of q can be obtained, except possibly for the last field K_{h}, as a subsequence of the function fields of the connected components V_{ν} of the k-varieties of ν-dimensional totally isotropic subspaces of q for ν = 0, …, [n/2]. For K_{h} we obtain a similar description, however the corresponding variety is defined over k(\sqrt{d(q)}). Hence each K_{i} is finitely and separably generated over k. The variety V_{ν} ×_{k} k_{s} is isomorphic to a quotient of G by some maximal proper parabolic subgroup P_{ν} of G, and its dimension equals the dimension of the unipotent radical of P_{ν} which is contained in the unipotent radical of some Borel subgroup. Let t_{i} denote the transcendence degree of K_{i}. Then t_{i} ≤ t, and hence, using a separating transcendence basis, we can embed K_{i} into a finite separable extension K′_{i} of K such that K′_{i} is a purely transcendental extension of K_{i}. Thus K_{i} and K′_{i} are k-specializations of each other and therefore the Witt indices of q_{K_{i}} and of q_{K′_{i}} are the same. This proves the theorem.

2.4 Remark. The theorem shows: Extending the field of definition of q by some purely transcendental field extension K of course does not increase the Witt index of q, but it does increase the magnitude of maximal subtori of SO_{q} to an extent which is required to produce a more subtle splitting behavior of q_{K} in the formation of separable extensions of K.

The theorem is not restricted to the case of characteristic ≠ 2. The proof holds in general, except that one has to replace the field k(\sqrt{d(q)}) by the corresponding field extension obtained by the Arf invariant.

The given bound is not optimal. As the proof shows it suffices to take for t the maximum of the dimensions of the unipotent radicals of maximal proper parabolic subgroups which are computed in [9, Cor. 5.5].

A theorem analogous to 2.3 holds in general for reductive groups. This will be discussed elsewhere.

3. On forms of height two.

3.0 Definition ([3, §1, p. 340]). A quadratic form q over k is called good if its leading form is defined over k.

We now assume that q is anisotropic.

3.1 Remark.

i) It follows from [11, 9.2 iii), p. 19] that, for a good form q over k, the leading form of q is always defined by a unique Pfister form over k.

ii) Let q be a good form of height 2 and odd dimension. Then q is known to be excellent (cf. [11, §10, p. 27]). This implies that dim q = 2^{r} −2^{s} +1 with 2 ≤ s < r, and the splitting pattern of q is given by (0, 2^{r−1}−2^{s}+1, 2^{r−1}−2^{s−1}) [5, 2.8 p. 187]. We refer also to [3, 1.2, p. 340].

In case of degree 1 we have:

3.2 Remark. Let q be of even dimension. If deg q = 1, then d(q) ≠ 1 and the leading form of q is defined by ⟨⟨−d(q)⟩⟩ (cf. [10, 5.10 ii), p. 82]), hence q is good. If in addition q is of height 2, then, by [11, 10.3, p. 28], we have the following two cases:

i) q is excellent. We then have q ≅ a⟨1, −b⟩ ⊗ ρ′ with a, b ∈ k^{∗} where ρ′ denotes the pure part of some Pfister form ρ; that is, ρ′ is the orthogonal complement of the subform ⟨1⟩ of ρ. In this case dim q = 2^{r} − 2 with r ≥ 3, and the splitting pattern of q is given by (0, 2^{r−1} − 2, 2^{r−1} − 1). ii) q is not excellent. Then we have dim q = 4, and the splitting pattern of q is given by (0, 1, 2).

We now assume that q is of height 2. In case of degree 2 we have for good forms:

3.3 Remark. Let q be good of height 2 and even dimension. If deg q = 2, then by [3, Thm. 1.6, p. 342] either q is excellent with dim q = 2^{r} − 4, r ≥ 4 and splitting pattern (0, 2^{r−1} −4, 2^{r−1} −2), or q is not excellent, q ≅ q_{1} ⊗⟨⟨a⟩⟩ with dim q_{1} = 4, which implies dim q = 8 and splitting pattern (0, 2, 4).

Forms q of height and degree 2 that are not good are treated in 3.11. For good forms of degree 3 we will obtain, as a special case of Theorem 3.9 below, the following natural generalization of the above remarks.

3.4 Theorem. Let q be good of height 2 with even dimension. If deg q = 3, then either q is excellent, dim q = 2^{r} − 8 with r ≥ 5 and splitting pattern (0, 2^{r−1} − 8, 2^{r−1} − 4), or q is not excellent, q_{K} ≅ q_{1} ⊗ ⟨⟨a, b⟩⟩ is anisotropic of height 2 over some field extension K of k with dim q_{1} = 4, hence dim q = 16, and splitting pattern (0, 4, 8).

Remark. In fact in [8, Thm. 2.12] it is shown that, for the non-excellent case, q ≅ q_{1} ⊗ ⟨⟨a, b⟩⟩ holds already over k.

We denote by I(k) the fundamental ideal of even dimensional form classes in the Witt ring of k.

3.5 Definition. Let n ∈ ℕ. We say that property E_{n} holds if and only if for every field k (of characteristic ≠ 2) the following conditions i) and ii) are satisfied:

i) For every i ∈ {1, …, n + 1} the map

⟨⟨−a_{1}, −a_{2}, …, −a_{i}⟩⟩ ↦ (a_{1}) ∪ (a_{2}) ∪ … ∪ (a_{i})

extends to a homomorphism e_{i} : I^{i}(k) → H^{i}(k, ℤ/2ℤ) with kernel I^{i+1}(k). ii) For every i ∈ {1, …, n} and for every i-fold Pfister form τ over k we have an equality

ker(H^{i+1}(k, ℤ/2ℤ) → H^{i+1}(k(τ), ℤ/2ℤ)) = e_{i}(τ) ∪ H^{1}(k, ℤ/2ℤ).

3.6 Remark. i) e_{0} is the parity of the dimension, e_{1} is given by the discriminant, e_{2} by the Clifford invariant. ii) Arason [1, 5.7, p. 490] resp. Jacob/Rost [6, Main Theorem, p. 552] have shown that the homomorphisms e_{3} resp. e_{4} exists and that properties E_{2} resp. E_{3} hold.

The content of the following lemma can essentially be found in [11, §10, p. 29-30], but, for convenience, we will give a proof here.

3.7 Lemma. Let q be an anisotropic form with dim q = 2^{N} for some N ≥ 2, h(q) = 2 and deg q = n < N and assume that q is good with leading form defined over k by τ. Then there is a field extension K of k such that q_{K} is anisotropic, h(q_{K}) = 2 and q_{K} and τ_{K} are divisible by an (n − 1)-fold Pfister form over K.

Proof. Let μ be a Pfister divisor of q of maximal degree r. Then if r > 0, by [11, 9.7, p. 23], τ splits over k(μ) and hence μ divides τ. By [11, 10.1, p. 28], τ does not divide q and hence r ≤ n − 1.

If r = n − 1 we are done. Hence let us assume that r < n − 1. After replacing q by some scalar multiple we may assume that q represents 1 over k. It follows from [11, 10.5, p. 29] that q ⊥ (−τ) has index dim μ, and that we can write

(∗)  q ⊥ (−τ) ≅ μ ⊗ ⟨1, −1⟩ ⊥ α

where α denotes the anisotropic kernel of q ⊥ (−τ).

Then dim q = 2^{N} < 2^{N} + 2^{n} − 2^{r+1} = dim α, and it follows from 1.12 that q_{k(α)} is anisotropic and h(q_{k(α)}) = 2. Clearly the leading form of q_{k(α)} is τ_{k(α)}.

Tensoring (∗) by k(α) we find that the index of q_{k(α)} ⊥ (−τ_{k(α)}) is bigger than that of q ⊥ (−τ), hence we obtain from [11, 10.5, p. 29] that q_{k(α)} has a Pfister divisor of degree bigger than r, but ≤ n − 1. Repeating this procedure, if necessary, yields the lemma.

3.8 Lemma. For n ≥ 1 let q = ψ⊗φ with an (n − 1)-fold Pfister form φ and an even dimensional regular form ψ. Then the following holds.

i) ψ ⊥ −⟨⟨−dψ⟩⟩ ∈ I^{2}(k) and q ∈ I^{n}(k), and the forms (ψ ⊥ −⟨⟨−dψ⟩⟩)⊗φ ∈ I^{n+1}(k) and ⟨⟨−dψ⟩⟩⊗φ ∈ I^{n}(k) depend on q only. ii) Assume that e_{i} : I^{i}(k) → H^{i}(k, ℤ/2ℤ) is defined for all i = 0, …, n + 1. Then the cohomology class

ẽ_{n+1}(q) := e_{n+1}((ψ ⊥ −⟨⟨−dψ⟩⟩)⊗φ) ∈ H^{n+1}(k, ℤ/2ℤ)

depends on q only. If q ∈ I^{n+1}(k), then ẽ_{n+1}(q) = e_{n+1}(q). If π is some n-fold Pfister form and a ∈ k^{∗} we have ẽ_{n+1}(aπ) = e_{n+1}(−⟨⟨−a⟩⟩⊗π).

Proof. i) The form ψ ⊥ −⟨⟨−dψ⟩⟩ has even dimension and discriminant 1. It is therefore in I^{2}(k) by [13, Ch. II, 2.2, p. 40]. After tensoring it with φ we obtain a form in I^{n+1}(k). Since in the Witt ring of k we have

q ⊥ (−ψ ⊥ ⟨⟨−dψ⟩⟩)⊗φ ∼ ⟨⟨−dψ⟩⟩⊗φ,

this shows the congruence

(∗)  q ≡ ⟨⟨−dψ⟩⟩⊗φ mod I^{n+1}(k),

hence ⟨⟨−dψ⟩⟩⊗φ ∈ I^{n}(k). If q = ψ′⊗φ′ is another decomposition of q with an (n − 1)- fold Pfister form φ′, then ψ′ is also even dimensional. By the above we obtain

⟨⟨−dψ⟩⟩⊗φ ⊥ −⟨⟨−dψ′⟩⟩⊗φ′ ≡ q ⊥ −q ∼ 0 mod I^{n+1}(k).

Since the left hand side is a difference of two n-fold Pfister forms and hence isotropic of dimension 2^{n+1} and the right hand side is in I^{n+1}(k) it follows from the Arason-Pfister Hauptsatz [13, 3.1, p. 289] that the left hand side is trivial in the Witt ring, hence ⟨⟨−dψ⟩⟩⊗φ = ⟨⟨−dψ′⟩⟩⊗φ′. This proves i).

ii) Clearly ẽ_{n+1}(q) depends only on q by i).

If q ∈ I^{n+1}(k) then, by (∗), it follows that ⟨⟨−dψ⟩⟩⊗φ ∈ I^{n+1}(k), and this form must be split by the Arason-Pfister Hauptsatz. Hence ẽ_{n+1}(q) = e_{n+1}(q).

We write the n-fold Pfister form π = ⟨⟨−b⟩⟩⊗φ with some (n − 1)-fold Pfister form φ and obtain from the above

\begin{aligned}ẽ_{n+1}(aπ) & = e_{n+1}((a⟨⟨−b⟩⟩ ⊥ −⟨⟨−b⟩⟩)⊗φ) \\ & = e_{n+1}(−⟨⟨−a⟩⟩⊗π). \end{aligned}

3.9 Theorem. Assume that property E_{n} (from 3.5) holds. Let q be a good form of height 2 and even dimension. If deg q = n, then either q is excellent, dim q = 2^{r} − 2^{n} with r ≥ n + 2 and splitting pattern (0, 2^{r−1} − 2^{n}, 2^{r−1} − 2^{n−1}) or q is not excellent, q_{K} ≅ q_{1} ⊗⟨⟨a_{1}, …, a_{n−1}⟩⟩ is anisotropic over some field extension K of k with dim q_{1} = 4, which implies dim q = 2^{n+1}, and splitting pattern (0, 2^{n−1}, 2^{n}).

For n = 2, this theorem was communicated to us by B. Kahn [7, 8], see also [3, Thm. 1.6, p. 342].

In the special cases n = 1, 2, 3 the reader will recognize the relationship to our statements in 3.2, 3.3 and 3.4.

The standard non-excellent example for 3.9 which is in generic position is given by the form

q = ⟨X_{1}, X_{2}, X_{3}, X_{4}⟩ ⊗ ⟨⟨Y_{1}, Y_{2}, … Y_{n−1}⟩⟩

over k = ℚ(X_{1}, …, X_{4}, Y_{1}, …, Y_{n−1}), say. The form q is anisotropic over k, since otherwise ⟨⟨1, 1⟩⟩⊗⟨⟨1, …, 1⟩⟩ would not be anisotropic over ℚ. Over K := k(\sqrt{−X_{1}X_{2}}), we have

q_{K} ≅ (⟨1, −1⟩ ⊥ ⟨a, −aX_{3}X_{4}⟩) ⊗ ⟨⟨Y_{1}, …, Y_{n−1}⟩⟩
≅ 2^{n−1} × ⟨1, −1⟩ ⊥ a⟨1, −X_{3}X_{4}⟩ ⊗ ⟨⟨Y_{1}, …, Y_{n−1}⟩⟩
≅ 2^{n−1} × ⟨1, −1⟩ ⊥ a⟨⟨−X_{3}X_{4}, Y_{1}, …, Y_{n−1}⟩⟩

with the last orthogonal summand being anisotropic over K. Thus we have i(q_{K}) = 2^{n−1}. Hence h(q) > 1 and q is not similar to a Pfister form, so, q is not excellent.

If, for some field extension F of k, the form q_{F} is isotropic, then q_{F} ≅ q_{1} ⊗ Θ with dim q_{1} = 4, Θ = ⟨⟨Y_{1}, …, Y_{n−1}⟩⟩, and q_{1} isotropic over F (cf. [2, Thm. 1.4, proof, p. 185]). So q_{1} ≅ ⟨1, −1⟩ ⊥ ⟨b, c⟩ and q_{F} ≅ 2^{n−1} × ⟨1, −1⟩ ⊥ ⟨b, c⟩ ⊗ Θ with ⟨b, c⟩ ⊗ Θ being similar to a Pfister form. If ⟨b, c⟩⊗Θ is anisotropic over F we have i(q_{F}) = 2^{n−1}. Otherwise it will split completely and i(q_{F}) = 2^{n}. In any case we find h(q) ≤ 2.

Therefore, h(q) = 2 with splitting pattern (0, 2^{n−1}, 2^{n}).

Proof of Thm. 3.9. Everything is clear if q is excellent, since we know all dimensions and splitting patterns of excellent forms of any height and degree, by [5].

Thus we assume that q is not excellent and that the leading form of q is defined by an n-fold Pfister form τ over k.

Step 1. q_{k(τ)} is similar to a Pfister form of degree ≥ n + 1. This follows from [11, 10.1 ii), p. 28]. Hence we know that dim q = 2^{r} for some r ≥ n + 1 and we want to show that r = n + 1. In other words, we know that q_{k(τ)} ∈ I^{n+1}(k(τ)) and we want to show that q_{k(τ)} ∉ I^{n+2}(k(τ)).

Step 2. By 3.7, there is a field extension K of k such that q_{K} is anisotropic and divisible by an (n − 1)-fold Pfister form, and q_{K} is still of height 2.

Thus we might as well assume that already over k we have q = q_{1}⊗Θ for some (n − 1)-fold Pfister form Θ and some form q_{1} whose dimension is an even power of 2.

Step 3. By 3.8 we obtain ẽ_{n+1}(q) ∈ H^{n+1}(k, ℤ/2ℤ).

Assume that q_{k(τ)} ∈ I^{n+2}(k(τ)), which means, by E_{n} (3.5 i)), that e_{n+1}(q_{k(τ)}) = 0. By E_{n} (3.5 ii)) and 3.8 we obtain for some a ∈ k^{∗}

(1)  ẽ_{n+1}(q) = e_{n}(τ) ∪ (a) = e_{n+1}(⟨⟨−a⟩⟩⊗τ).

Since τ_{k(q)} is the leading form of q, there is an f ∈ k(q)^{∗} such that (q_{k(q)})_{an} = fτ_{k(q)}, and we obtain using 3.8 ii)

(2)  ẽ_{n+1}(q_{k(q)}) = e_{n+1}(−⟨⟨−f⟩⟩⊗τ_{k(q)}).

Let ρ := fτ_{k(q)} ⊥ −aτ_{k(q)}. Then, in the Witt ring of k(q), we have an equivalence

ρ ∼ (−⟨⟨−f⟩⟩ ⊥ ⟨⟨−a⟩⟩_{k(q)})⊗τ_{k(q)},

hence we conclude that ρ ∈ I^{n+1}(k(q)), and equations (1), (2) yield e_{n+1}(ρ) = 0. It now follows from E_{n} (3.5 i)) that ρ ∈ I^{n+2}(k(q)). Since dim ρ = 2^{n+1}, the Arason-Pfister Hauptsatz [13, Ch. X, 3.1, p. 289] tells us that ρ splits; in other terms, aτ_{k(q)} ≅ fτ_{k(q)}.

Since a ∈ k^{∗}, we conclude that also fτ_{k(q)} is defined over k; that is, the anisotropic kernel of q over k(q), and hence all higher kernels of q, are defined over k. This means that q is excellent [11, 7.14, p. 6], which is a contradiction.

So, indeed, dim q = 2^{n+1} and dim q_{1} = 4.

Step 4. It only remains to determine the splitting pattern of an anisotropic form q = q_{1}⊗Θ of height 2 with dim q_{1} = 4 and a Pfister form Θ.

It follows from [2, Thm. 1.4, p. 185] that if q_{1}⊗Θ is isotropic, then there is an isotropic form q′_{1} with q_{1}⊗Θ ≅ q′_{1}⊗Θ. Thus i_{1}(q) ≥ dim Θ, which in our case yields i_{1}(q) = 2^{n−1}; that is, q has splitting pattern (0, 2^{n−1}, 2^{n}) (cf. the standard example above).

3.10 Corollary. If E_{n} holds, then there are no anisotropic forms q of height 2 and degree n with leading form defined over the base field and dim q = 2^{N} with N ≥ n + 2.

Hence if the condition E_{n} holds, then the answer to the question 10.6 posed by M. Knebusch [11, 10.6, p. 30] is negative. In particular, 3.6 ii) shows that the answer is negative for n = 2, 3.

Let us now discuss the situation when q is of height 2 but not good. Then q is not excellent, hence if dim q is odd, then dim q ≥ 11 by 1.16. If dim q is even, then deg q ≥ 2 by 3.2, in particular dim q ≥ 6.

3.11 Theorem. Let q be an anisotropic form of height 2 which is of even dimension and not good. If deg q = 2, then q is of dimension 6 and has splitting pattern (0, 1, 3).

This theorem is due to B. Kahn [8, 2.11]. We will give here a proof based on 1.6.

Proof. As in 1.6, we have E(q) = M_{2^{r}}(D) for some skew field D, and it follows from the assumption on the degree that D ≠ k and that d(q) = 1. If D were a quaternion skew field, then the leading form of q would be defined over k by [11, 9.8, p. 23]. Hence D is the product of at least two quaternion skew fields.

Since the height is 2 we are in case ii) of 1.6, hence dim q = 6 and the splitting pattern is (0, 1, 3).

3.12 Remark. In [3, Prop. 2.8, p. 348] it was proved that forms as in 3.11 have a dimension ≤ 26.

3.13 Remark. The standard example of a (6-dimensional) quadratic form of height 2 and degree 2 that is not good is given by

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

over k = F (X_{1}, X_{2}, X_{3}, X_{4}, X_{5}); see Prop. 1.2.

References.

[1] J. Kr. Arason, Cohomologische Invarianten quadratischer Formen, J. Algebra 36 (1975), 448-491.

[2] R. Elman and T. Y. Lam, Pfister forms and K-Theory of fields J. Algebra 23 (1972), 181-213.

[3] R. W. Fitzgerald, Quadratic forms of height two, Transactions AMS 283 (1984), 339-351.

[4] D. W. Hoffmann, Isotropy of quadratic forms over the function field of a quadric, preprint (1994), to appear in Math. Z.

[5] J. Hurrelbrink and U. Rehmann, Splitting patterns of excellent quadratic forms, J. reine angew. Math. 444 (1993), 183-192.

[6] B. Jacob and M. Rost, Degree four cohomological invariants for quadratic forms, Invent. math. 96 (1989), 551-570.

[7] B. Kahn, Oral communication at K-Theory conference at Oberwolfach, June 1993.

[8] B. Kahn, Formes quadratiques de hauteur et de degré 2, preprint (1993), to appear.

[9] I. Kersten and U. Rehmann, Generic splitting of reductive groups, Tôhoku Math. J., 46 (1994), 35-70.

[10] M. Knebusch, Generic splitting of quadratic forms, I, Proc. London Math. Soc. 33 (1976), 65-93.

[11] M. Knebusch, Generic splitting of quadratic forms, II, Proc. London Math. Soc. 34 (1977), 1-31.

[12] M. Knebusch and W. Scharlau, Algebraic theory of quadratic forms, DMV-Seminar 1, Birkhäuser Boston-Basel-Stuttgart, 1980.

[13] T. Y. Lam, The algebraic theory of quadratic forms, Math. Lecture Notes Ser. Benjamin, Reading, Mass., 1973.

[14] A. S. Merkurjev, Simple algebras and quadratic forms, Izv. Akad. Nauk SSSR, Ser. Mat. 55 (1991), 218-224; English trans: Math. USSR Izvestiya 38 (1992), 215-221.

[15] A. Pfister, Quadratische Formen in beliebigen Körpern, Invent. math. 1 (1966), 116-132.

Jürgen Hurrelbrink  Ulf Rehmann
Department of Mathematics  Fakultät für Mathematik
Louisiana State University  Universität Bielefeld
Baton Rouge, La. 70803  33501 Bielefeld
USA  Germany