Splitting Patterns of Excellent Quadratic Forms

(to appear in Crelle’s Journal)

Jürgen Hurrelbrink* and Ulf Rehmann

Dedicated to Martin Kneser on the occasion of his 65^{th} birthday

Abstract. The Witt indices which may occur after base field extension for a given anisotropic excellent quadratic form are uniquely determined by the dimension of the form and independent of the base field and of the particular form chosen. We generalize that result to a larger class of forms and give an easy method of computing these indices. In particular, this can be done for the form given by a sum of squares. This gives a natural generalization of Pfister’s theorems on the level of fields. Another result is the existence of families of anisotropic special orthogonal groups of unbounded absolute rank which have a joint generic splitting field of small transcendence degree including examples of transcendence degree 0.

Introduction

The splitting pattern of a regular quadratic form q over a field k of characteristic different from 2 is the sequence of distinct Witt indices of q which may occur after base field extension. For a given regular form q̄ we define a class of forms which we call q̄-extensions. We will show that the anisotropic members q of this class all have a splitting pattern depending only on the dimension of q and on the splitting pattern of q̄. This dependency is determined in §2. For example, Pfister forms are of this type (here we have q̄ = 0); their Witt index is either 0 or maximal. For every form q̄ there are q̄-extensions of arbitrarily high dimension. The q̄-extensions with dim q̄ ≤ 1 are precisely the “excellent” forms introduced by Knebusch, who gave a recursive formula for their possible Witt indices in [3, 7.11]. In general, the determination of the splitting pattern of a given quadratic form seems to be very difficult, however, for excellent forms there is an easy way of describing their splitting patterns; this will be discussed in §2.

Our investigations have interesting consequences. For example, they generalize the results of Pfister that the level of fields is infinite or a power of 2, and that fields exist having a prescribed power of 2 as its level (cf. 3.4).

Another important consequence is the fact that families of anisotropic forms of arbitrarily high dimension exist which have a field of low transcendence degree as a joint generic splitting field (cf. 2.5, 2.14). This behavior is completely different from the splitting behavior of (finite-dimensional) central simple division algebras over k, since a theorem of Witt, Amitsur and Roquette says that a generic splitting field of such an algebra D splits precisely those division algebras which represent powers of D in the Brauer group. This is always a finite set of division algebras.

1. Generic splitting towers of quadratic forms Let V be a vector space of dimension n ≥ 2 over a field k of characteristic different from 2 and q : V → k be a regular quadratic form on V with discriminant d(q) ∈ k^{∗}/k^{∗2}. We are interested in the possible Witt indices

0 ≤ i_{0} < i_{1} < … < i_{h−1} < i_{h} = [n/2]

* The first-named author gratefully acknowledges support by the Sonderforschungsbereich 343 “Diskrete Strukturen in der Mathematik” during a stay at the University of Bielefeld in May 1992

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

These Witt indices can be investigated by means of the notion of the so-called generic splitting tower of q considered by Knebusch [2, §5] (cf. also [4, Ch. 4, 6.9, p. 160]). However, our definition is slightly more general than that of [2].

For any quadratic form q, let i(q) denote its Witt index.

1.1 Definition. A generic splitting tower of q is a sequence k = K_{0}, K_{1}, …, K_{h} of field extensions of k with the following properties:

i) The associated sequence of Witt indices i_{j} := i(q_{K_{j}}) is strictly increasing. ii) For every field extension L of k, there is a j such that i(q_{L}) = i_{j}. iii) Every field extension L of k with i(q_{L}) ≥ i_{j} is a k-specialization of K_{j} (that is,

there is a k-place K_{j} → L ∪ {∞}). The sequence (i_{0}, i_{1}, …, i_{h}) is called the splitting pattern of q, the number h is called the height of q and will be denoted by h(q).

1.2 Remark. i) and iii) imply the existence of the k-specializations K_{i} → K_{j} for i < j. Hence we obtain a sequence k = K_{0} → K_{1} → … → K_{h} of k-specializations. In [2] a sequence of embeddings was required. But our definition allows generic splitting towers {K_{i}} with not necessarily non-decreasing transcendence degrees, which often exist, as can easily be seen from 2.5, 2.10, 2.14.

If {k = L_{0}, L_{1}, …, L_{h′}} is a second generic splitting tower of q, then obviously h′ = h and for j = 0, …, h the fields K_{j} and L_{j} are k-spezializations of each other.

A natural choice for a generic splitting tower of q is described in [1, §5]: For i = 1, …, [n/2], the variety of i-dimensional totally isotropic subspaces of V is defined over k except if n is even, i = n/2 and d(q) ≠ 1, in which case it is defined over k(\sqrt{d(q)}). (This variety is absolutely irreducible except when n is even and i ≥ n/2−1. In this case it consists of two irreducible components, and for our purposes we just may take one of them, cf. [1, 5.2].) Its function field F_{i} is a generic field for the problem of splitting off at least i hyperbolic planes from V. That is, i(q_{F_{i}}) ≥ i, and every field extension L of k with i(q_{L}) ≥ i is a k-specialization of F_{i}. Also, for every such L, we have i(q_{L}) ≥ i if and only if the free composite F_{i} · L is a purely transcendental extension of L [1, 5.3 and 5.7]. We get a generic splitting tower of q by taking the subsequence of {F_{i}} obtained by avoiding repetitions of Witt indices; that is, we define K_{0} = k, and, if K_{j} is already defined and i_{j} < [n/2], then K_{j+1} is F_{i} where i is minimal with i > i_{j}. If i_{j} = [n/2] we set h = j (cf. [1, 5.8 and 5.9]). If necessary, we write F_{i}(q) (resp. K_{i}(q)) instead of F_{i} (resp. K_{i}).

It is immediate from 1.1 ii) that the numbers i_{j} are the only possible Witt indices for q_{L}, where L runs through all field extensions of K.

For example, if q is a regular subform of codimension ≤ 1 of an anisotropic Pfister form, then the only possible Witt indices of q are i_{0} = 0 and i_{1} = [n/2] since, for every such form q and every field extension L of k, we have: if q_{L} is isotropic, then q_{L} splits completely. For Pfister forms this follows from [4, Ch. 4, Cor. 1.5, p. 144]. If q is of codimension 1 in some Pfister form, say φ = q ⊥ ⟨a⟩ for some a ∈ k^{∗}, and if q_{L} is isotropic, then φ_{L} splits totally and therefore has an (n + 1)/2-dimensional totally isotropic subspace which intersects the space of q_{L} in a subspace of dimension [n/2], hence q_{L} splits completely.

2. Excellent quadratic forms

2.1 Remark. Let q̄ = ⟨a_{1}, …, a_{n}⟩ be a regular quadratic form over k. Then the Pfister form φ := ⟨1, a_{1}^{−1}a_{2}⟩ ⊗ … ⊗ ⟨1, a_{1}^{−1}a_{n}⟩ is of dimension 2^{n−1} and has an orthogonal summand ⟨1, a_{1}^{−1}a_{2}, …, a_{1}^{−1}a_{n}⟩ which is similar to q̄. Thus we have an orthogonal decomposition a_{1}φ ≅ q ⊥ q̄ with some regular form q.

2.2 Definition. Let q be a regular quadratic form over k.

i) The level ℓ(q) of q is the smallest number l such that q is an orthogonal summand of a scalar multiple of an anisotropic l-dimensional Pfister form.

ii) q is a Pfister neighbour if there exists a Pfister form φ with (dim φ)/2 < dim q, some a ∈ k^{∗} and some form q′ such that aφ ≅ q ⊥ q′. The forms φ (resp. q′) are called the associated Pfister form (resp. the complementary form) of q, and dim q′ is called the codimension of q.

iii) Let q̄ be a regular quadratic form over k. The form q is called a q̄-extension if there is a sequence q = q_{0}, q_{1}, …, q_{r} = q̄ of quadratic forms over k of length r ≥ 0 such that, for j < r, the form q_{j} is a Pfister neighbour with complementary form q_{j+1}. The number r is called the order of the q̄-extension q and denoted by ord (q, q̄). An excellent form q is a q̄-extension with dim q̄ ≤ 1, and ord (q) denotes its order.

2.3 Remarks and Examples.

i) The notions of a Pfister neighbour and of an excellent form have been introduced and studied by Knebusch [3, 7.4ff.]. It is proved in [3, l.c.] that φ and q′ in 2.2 ii) are uniquely determined up to isomorphism by q, hence it follows that the sequence q_{0}, …, q_{r} in 2.2 iii) is uniquely determined up to isomorphism.

ii) Clearly the level of a form q is always a power of 2. By 2.1 we obtain ℓ(q) ≤

2^{(dim q)−1}. A form q is a Pfister neighbour if and only if ℓ(q) < 2 dim q. iii) Let n, t ∈ ℕ and let t be minimal such that 2^{t} ≥ n. Then the quadratic form

n × ⟨1⟩ is excellent of level l = 2^{t}, as is easily seen by induction. Hence there are (anisotropic) excellent forms of any dimension.

iv) If q̄ is excellent (which is in particular true if dim q̄ ≤ 3), then every q̄-extension

is also excellent.

v) For excellent anisotropic q, it follows from [3, Prop. 7.9] that h(q) = ord (q). In 2.12 below we will show that, for an arbitrary anisotropic q̄-extension q, the relation h(q) ≤ ord (q, q̄) + h(q̄) holds with equality for excellent forms, which shows that ord (q, q̄) plays the role of a ’relative’ height for anisotropic forms.

2.4 Proposition. Let q be a q̄-extension of order r. If q̄ is isotropic, then the anisotropic kernels of q and of (−1)^{r} q̄ are isomorphic. If q̄ is anisotropic, then the anisotropic kernel of q is isomorphic to one of its ’signed’ complementary forms (−1)^{j} q_{j} for some j with 0 ≤ j ≤ r. In particular, the anisotropic kernel of an excellent form is excellent.

Proof. We may assume that q is isotropic. Therefore q ⊥ q′ splits completely since it is similar to an isotropic Pfister form, and hence q defines the same Witt class as −q′, which gives the assertion.

We give an example concerning generic splitting fields of special orthogonal groups. Recall that a generic splitting field of a reductive linear algebraic k-group G in the sense of [1] is a field extension F of k such that G_{F} is split and such that the field extensions L of k for which G_{L} is split are the k-specializations of F. If G = SO(q) is the special orthogonal group of the regular quadratic form q of dimension n, then G_{F} is split if and only if i(q_{F}) = [n/2]. This condition means that q_{F} is totally split.

2.5 Theorem. Let q be a q̄-extension with dim q̄ ≥ 2, let G = SO(q) and Ḡ = SO(q̄). A field extension F of k is a (generic) splitting field of G if and only if it is a (generic) splitting field of Ḡ.

Proof. It follows immediately from 2.4 and the preceding remark that, for dim q̄ ≥ 2, the field F is a splitting field of G if and only if it is a splitting field of Ḡ.

Let F be a generic splitting field of Ḡ and L a splitting field of G. Then L is a splitting field of Ḡ and hence is a k-specialization of F. Therefore F is a generic splitting field of G. The converse follows similarly.

2.6 Lemma. Every natural number n can be written uniquely as an alternating sum of 2-powers

n = 2^{a_{h}} − 2^{a_{h−1}} + 2^{a_{h−2}} − + … + (−1)^{h−1}2^{a_{1}} + (−1)^{h}ϵ

with ϵ, h, a_{1}, …, a_{h} ∈ ℕ ∪ {0} satisfying 0 < a_{1} < a_{2} < … < a_{h−1} < a_{h} and

ϵ = \{ \begin{matrix}0 and a_{1} < a_{2} − 1 & if n is even, \\ 1 and 1 < a_{1} & if n is odd. \end{matrix}

In particular, 2^{a_{h}} is the smallest 2-power greater than or equal to n.

Proof. Let n = 2^{g}ϵ_{0} + 2^{g−1}ϵ_{1} + … + 2ϵ_{g−1} + ϵ_{g} denote the dyadic expansion of n with ϵ_{i} ∈ {0, 1}. Each subsequence of {ϵ_{i}} consisting of consecutive 1’s only and being maximal with this property represents some subsummand of type 2^{a} − 2^{b} with a > b. If the last one is such that a = b + 1, then this can be replaced by 2^{b}. This proves the Lemma.

2.7 Definition. For n ∈ ℕ, the expansion of 2.6 is called the alternating 2-expansion of n. For j = 0, …, h, the numbers 2^{a_{j}} − 2^{a_{j−1}} + − … ± 2^{a_{1}} ∓ ϵ are called the j-th alternating 2-partial sums of n and denoted by n^{(j)}. The number h is called the height h(n) of n.

2.8 Corollary. Let q be a q̄-extension of order r with dim q = n, dim q̄ = n_{0}, and let n be expanded as in 2.6. Then h = h(n) = r + h(n_{0}), ℓ(q) = 2^{a_{h}} and dim q_{j} = n^{(h−j)} for any j = 0, …, r. If q̄ is anisotropic and j is such that (−1)^{j}q_{j} is the anisotropic kernel of q, then i(q) is given by

i_{j}(n) := (n − n^{(h−j)})/2 = \{ \begin{matrix}2^{a_{h}−1} − + … − 2^{a_{h−j+1}−1} & if j is even, \\ 2^{a_{h}−1} − + … + 2^{a_{h−j+1}−1} − dim q_{j} & if j is odd. \end{matrix}

In particular, for j = 1, the smallest non-trivial Witt index is given by

\begin{aligned}i_{1}(n) & = 2^{a_{h}−1} − 2^{a_{h−1}} + 2^{a_{h−2}} − + … + (−1)^{h−1}2^{a_{1}} + (−1)^{h}ϵ \\ & = n − 2^{a_{h}−1} = n − ℓ(q)/2, \end{aligned}

which is the remainder of n modulo the highest power of 2 below n.

2.9 Proposition. Let q be an anisotropic q̄-extension with r = ord (q, q̄) > 0 and with complementary form q′.

i) Let r > 1 or let q′_{F_{1}(q)} be anisotropic. Then −q′_{F_{1}(q)} is isomorphic to the anisotropic kernel of q_{F_{1}(q)}.

ii) If q′_{F_{1}(q)} is isotropic, then the fields F_{1}(q) and F_{1}(q′) are k-specializations of each other.

Remark. It is not known whether the condition in 2.9 ii) can hold for a Pfister neighbour q with complementary form q′. (Cf. [3, 8.3].)

Proof. i) By 2.4 we only have to show that q′_{F_{1}(q)} is anisotropic if q′ is a Pfister neighbour. Hence we may assume that r > 1. Let φ′ be the Pfister form associated to q′. Remark 2.3.ii) implies dim q > dim φ′. If q′_{F_{1}(q)} were isotropic, then φ′_{F_{1}(q)} would split completely. Hence, by the Cassels-Pfister subform theorem [4, Ch. 4, 5.4 ii), p. 155], q would be a subform of φ′, which is impossible.

ii) Clearly F_{1}(q) is a k-specialization of F_{1}(q′). Conversely, F = F_{1}(q′) splits the Pfister form φ associated to q_{F} totally, and hence any of its maximal isotropic subspaces intersects the space of q_{F} non-trivially for dimension reasons, hence q_{F} is isotropic and therefore F is a k-specialization of F_{1}(q).

2.10 Corollary. Let 1 ≤ i ≤ [n/2].

i) For j = 1, …, r − 1 the anisotropic kernel of q_{F_{i}(q)} is isomorphic to (−1)^{j}(q_{j})_{F_{i}(q)} if and only if i_{j−1}(n) + 1 ≤ i ≤ i_{j}(n). This is also true for j = r if q̄_{F_{1}(q_{r−1})} is anisotropic.

ii) Let K_{0} = k and for j = 1, …, r let K_{j} = F_{1}(q_{j−1}). Then the sequence {K_{j}} is an initial sequence of a generic splitting tower of q. In particular, if q is excellent, then this is a generic splitting tower.

Proof. i) If 1 ≤ i ≤ i_{1}(n), then the free composite F of the fields F_{1}(q) and F_{i}(q) is a purely transcendental extension of F_{1}(q) by [1, Th. 5.3]. Therefore −q′ is anisotropic over F, since it is so over F_{1}(q) by 2.9, and hence it is anisotropic over F_{i}(q). It follows that the anisotropic kernels of q_{F_{1}(q)} and of q_{F_{i}(q)} are both obtained from −q′ by base extension. On the other hand, if i > i_{1}(n), then i(q_{F_{i}(q)}) ≥ i > i_{1}(n) and hence q′_{F_{i}(q)} is isotropic. This proves the claim for r = 1. Since the complementary form of q_{j} is q_{j+1}, an induction on j gives the rest of the statement.

ii) This follows from 2.8, applied to the form q_{K_{j}} over K_{j}.

2.11 Theorem. Let q be an anisotropic q̄-extension of order r = ord (q, q̄) with n = dim q. Then the first r components of the splitting pattern of q are always given by i_{j} = i_{j}(n) for j = 0, …, r − 1. Its components for j ≥ r are given by

i_{j} = i_{r}(n) + \{ \begin{matrix}j_{j−r} & if q̄_{F_{1}(q_{r−1})} is anisotropic, \\ j_{j−r+1} & otherwise, \end{matrix}

where (j_{0}, …, j_{t}) is the splitting pattern of q̄.

Proof. The j^{th} component of the splitting pattern of q is the Witt index of q_{K_{j}}. By 2.8, applied to q_{K_{j}} instead of q, this is i_{j}(n) for j < r, and the same argument, together with 2.9, works for j = r. For j > r the statement follows from 2.4, applied to q_{K_{ν}(q̄)}, where K_{ν}(q̄) runs through a generic splitting tower of q̄.

2.12 Corollary. If q is an anisotropic q̄-extension of order r, then h(q) ≤ ord (q, q̄) + h(q̄). Equality holds if q̄_{F_{1}(q_{r−1})} is anisotropic, and otherwise we have h(q) = ord (q, q̄)+ h(q̄) − 1. In particular, if q is excellent, then we have h(q) = h(dim q).

2.13 Illustration. Let q be an anisotropic excellent form of dimension n over a field k. The following data hold independently of the field k and of the particular excellent form q. For example, they hold for the form given by the sum of n squares over any field with level ≥ n.

i) n = 12 = 2^{4} − 2^{2}, thus h(q) = 2 and dim q_{1} = 4, dim q_{2} = 0. Splitting pattern: (0, 4, 6).

ii) n = 123 = 2^{7} − 2^{3} + 2^{2} − 1, thus h(q) = 3 and dim q_{1} = 5, dim q_{2} = 3, dim q_{3} = 1. Splitting pattern: (0, 59, 60, 61).

iii) n = 1234 = 2^{11} − 2^{10} + 2^{8} − 2^{6} + 2^{5} − 2^{4} + 2, thus h(q) = 7 and dim q_{1} = 814, dim q_{2} = 210, dim q_{3} = 46, dim q_{4} = 18, dim q_{5} = 14, dim q_{6} = 2, dim q_{7} = 0. Splitting pattern: (0, 210, 512, 594, 608, 610, 616, 617).

The anisotropic quadratic forms of height 1 are known to be the forms similar to an anisotropic subform of a Pfister form of codimension ≤ 1; that is, these are exactly the anisotropic excellent forms of dimension 2^{a_{1}} or 2^{a_{1}} − 1 (cf. [2, Th. 5.8]).

The anisotropic excellent forms of height 2 are clearly the anisotropic excellent forms of dimension 2^{a_{2}} − 2^{a_{1}} with a_{1} ≥ 1, a_{2} > a_{1} + 1 or of dimension 2^{a_{2}} − 2^{a_{1}} + 1 with a_{1} ≥ 2 and a_{2} > a_{1}. Compare [3, §10] for results and questions about anisotropic forms of height 2 that are not excellent.

2.14 Corollary. Let q be excellent of dimension n. Let n be expanded as in 2.6. Then G = SO(q) has a generic splitting field of transcendence degree ≤ 2^{a_{1}} − ϵ − 2. In particular, G has an algebraic generic splitting field of degree 2 – namely k(\sqrt{d(q)}) – if n ≡ 2 (mod 4), and G has a generic splitting field which is the function field of a complete curve defined over k if n ≡ 3, 4, 5 (mod 8). The curve can be taken to be the Severi-Brauer variety associated to the even Clifford algebra of the ternary form q_{h−1} in case n ≡ 3, 5 (mod 8). Similarly, in case n ≡ 4 (mod 8) the curve can be taken to be the Severi-Brauer variety associated to the quaternion algebra which is related to the quaternary form q_{h−1}.

Proof. q is a q_{h−1}-extension for the (excellent) form q_{h−1} of dimension n_{1} = 2^{a_{1}} − ϵ. Hence, by 2.5, a generic splitting field of SO(q_{h−1}) is one of G. By the preceding remark and by [1, 5.4], q_{h−1} is of height 1, hence the field F_{1}(q_{h−1}) (resp. k(\sqrt{d(q_{h−1})}) is a generic splitting field of SO(q_{h−1}) if n_{1} ≥ 3 (resp. n_{1} = 2). By [1, 5.5], this field has transcendence degree n_{1} −2. Let n ≡ 3, 5 (mod 8). It follows from [4, Ch. 2, 14.3 (i)], that the ternary form q_{h−1} splits if and only if its even Clifford algebra C = C(q_{h−1}) splits. But C is a quaternion algebra, and it has been discovered by Witt [5] a long time ago that the generic splitting field of a quaternion algebra is the function field of its associated Severi-Brauer variety; that is, of the variety of its nilpotent elements (cf. [1, 3.20]).

The curve in case n ≡ 4 (mod 8) is obtained as the Severi-Brauer variety of the quaternion algebra A associated to the Pfister form related to q_{h−1}. Namely, since in this case d(q_{h−1}) = 1, it follows from [4, Ch. 2, 14.3 (ii)] that q_{h−1} splits if and only if A splits.

3. Witt indices, the level of fields, and applications to number fields. Corollary 2.8 gives some interesting relations between the possible Witt indices and the height of excellent quadratic forms. For anisotropic excellent forms q of height 1 everything is clear: i_{1} = [n/2] with n = dim q. In the case of height ≥ 2 we obtain:

3.1 Proposition. Let q be an anisotropic excellent form of height h ≥ 2. Then i_{1}(n) ≥ 2^{h−2} with strict inequality for h ≥ 4.

Proof. For h = 2 there is nothing to prove. Hence we may assume h ≥ 3. Let n be given in alternating 2-expansion as in 2.6. It follows from 2.8 that

i_{1}(n) = 2^{a_{h}−1} − 2^{a_{h−1}} + dim q_{h−2} = 2^{a_{h−1}}(2^{a_{h}−a_{h−1}−1} − 1) + dim q_{h−2} ≥ dim q_{h−2}.

Hence we have i_{1}(n) ≥ dim q_{h−2} > ℓ(q_{h−2})/2 = 2^{a_{h−2}−1}, where the inequality follows immediately from the definition of the level of a quadratic form, and since a_{j} − 1 ≥ j for every j = 2, …, h, we obtain in fact i_{1}(n) > 2^{h−2} for h ≥ 4.

The information about the dimensions of the anisotropic kernels q_{j} stated in 2.8 makes the parity of all Witt indices explicit for anisotropic excellent forms of dimension n. One concludes:

Let n be even; then all Witt indices are even, except – of course – possibly the last one, i_{h} = [n/2].

Let n be odd; then the parity of the Witt indices is alternating; namely, i_{j} ≡ j (mod 2) for 0 ≤ j ≤ h.

In particular, this yields for odd dimensional forms:

3.2 Remark. An anisotropic excellent form of odd dimension n has an even height if and only if n ≡ 1 (mod 4).

Proof. For anisotropic excellent forms of odd dimension n we have n = 2 i_{h} + 1 with i_{h} ≡ h (mod 2).

An immediate consequence of the last statement in 2.8 is the following corollary.

3.3 Corollary. An isotropic excellent form q is of Witt index 1 if and only if its (first) complementary form is anisotropic and dim q = 2^{κ} + 1 for some κ ∈ ℕ ∪ {0}.

3.4 Remark. Of course 3.3 implies, in the case of sums of squares, Pfister’s theorem that the level of any field k is either infinite or a power of 2 (cf. [4, Ch. 2, 10.8, p. 71]). Namely, the level s = s(k) of k is the smallest number s such that Q_{s} := s × ⟨1⟩ represents −1 over k, or, equivalently, the biggest s such that Q_{s} is anisotropic over k. It follows from 2.3 iii) and 3.3, applied to Q_{s+1}, that this number is infinite or a power of 2, which is Pfister’s result. If s(k) is finite, the anisotropic form Q_{s(k)} is a Pfister form and hence its level is just the level of k.

The statements 2.9 and 3.3 also give a proof of the fact that fields of arbitrary level exist (cf. [4, Ch. 4, 4.3 , p. 152]): Let q = (2^{κ} + 1) × ⟨1⟩ over some formally real field k. Then, by 2.9 and 3.3, the field F_{1}(q) is of level 2^{κ}. In fact, if k is taken to be ℚ, the field of rationals, then this gives a “generic construction” of such a field.

As another application, we describe the splitting behavior of excellent forms over ℚ after base extension by number fields F.

3.5 Theorem. Let q be an anisotropic excellent form over ℚ of dimension n. Then, for every number field F, the only possible Witt indices of q_{F} occur among

0, [n/2]  if n ≡ 0, ±1 (mod 8),
0, [n/2] − 1, [n/2] if n ≡ ±2, ±3 (mod 8),
0, [n/2] − 2, [n/2] if n ≡ 4  (mod 8).

Proof. For n ≤ 3 our claim is obvious; for n = 4 it is true since the splitting pattern is given by i_{0} = 0, i_{1} = 2. Hence we may assume that q is of dimension n ≥ 5. Let F be a number field such that q_{F} is isotropic.

Since every 5-dimensional indefinite form q over ℚ is isotropic, we know that q is definite; q = ⟨x_{1}, …, x_{n}⟩ with x_{i} ∈ ℚ^{∗}, all x_{i} being positive or all x_{i} being negative. If the number field F had a real embedding, then q_{F} would still be anisotropic. So, F is totally imaginary and hence, by Minkowski-Hasse, the anisotropic kernel of q_{F} is of dimension ≤ 4. Thus, the Witt index i = i(q_{F}) satisfies i ≥ (n − 4)/2 ≥ [n/2] − 2. We expand n as in Lemma 2.6. It follows immediately from 2.6 that a_{2} ≥ 3 and hence

n ≡ (−1)^{h−1}n^{(1)} = (−1)^{h−1}(2^{a_{1}} − ϵ) (mod 8).

If n ≡ 0, ±1 (mod 8), then a_{1} ≥ 3 and hence i = [n/2]. If n ≡ ±2 (mod 8), then ϵ = 0, a_{1} = 1 and hence i = [n/2] or [n/2] − 1. If n ≡ ±3 (mod 8), then ϵ = 1, a_{1} = 2 and hence also i = [n/2] or [n/2] − 1. If n ≡ 4 (mod 8), then ϵ = 0, a_{1} = 2 and hence i = [n/2] or [n/2] − 2.

It is straightforward to extract from the statement of 3.5 the well-known fact that the level of number fields, if finite, is 1,2 or 4.

3.6 Remark. The indices mentioned in 3.5 do occur over number fields. To see this, let F be a number field of level 4. Put Q_{n} = n×⟨1⟩. Then e.g. i((Q_{6})_{F}) = 2 = [6/2]−1 and i((Q_{12})_{F}) = 4 = [12/2] − 2 and similarly for other values of n.

Proof. Clearly (Q_{n})_{F} is a (Q_{j})_{F} -extension for some j ≤ 4. Now one applies 2.4.

References

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

[4] W. Scharlau, Quadratic and Hermitian forms, Grundlehren der math. Wissenschaften, 270, Springer Berlin-Heidelberg-New York, 1985.

[5] E. Witt, Über ein Gegenbeispiel zum Normensatz, Math. Z. 39 (1935), 462-467.

Jürgen Hurrelbrink  Ulf Rehmann
Department of Mathematics  Fakultät für Mathematik
Louisiana State University  Universität Bielefeld
Baton Rouge, La. 70803  4800 Bielefeld 1
USA  Germany