Splitting Patterns and Trace Forms

Jurgen Hurrelbrink* and Ulf Rehmann

Published in Canad. Math. Bull. 41 (1998), no. 1, 71–78

Abstract: The splitting pattern of a quadratic form q over a field k consists of all distinct Witt indices that occur for q over extension fields of k. In small dimensions, the complete list of splitting patterns of quadratic forms is known. We show that all splitting patterns of quadratic forms of dimension at most nine can be realized by trace forms. 1991 Mathematical Subject Classification: 11E04. Keywords and Phrases: Quadratic forms, Witt indices, generic splitting.

0. Introduction The history of trace forms dates back to the middle of the nineteenth century. At that time, Sylvester, Jacobi, and Hermite determined the number of real roots of polynomials in one variable over R in terms of – as we would say today – the signature of trace forms. This is nicely explained in the excellent article on “Galois cohomology and the trace form” by E. Bayer-Fluckiger [1], p. 36.

Splitting patterns (cf. [8]) are invariants of the similarity classes of quadratic forms. Hence splittinq patterns of trace forms are invariants of finite separable field extensions and therefore of particular interest: here is an illustration of information they provide for example about the level of the normal closure of field extensions.

Let k be a field of characteristic ≠ 2. Let the quadratic form q over k be the trace form of a separable field extension F of k of degree n. Let N be any field containing a normal closure of F over k. We have:

(1) If the Witt index i of q over k is positive, then N has finite level s(N) ≤ n − i.

(2) If q is not stably excellent, then N has finite level s(N) ≤ n − 1.

The first statement follows from Witt cancellation and the result that, over N, the trace form q is isometric to a sum of n squares, [2]. The second statement is a corollary since, over N, trace forms that are not stably excellent will be isotropic, [9].

1. Constraints on splitting patterns Let q be an anisotropic quadratic form of dimension n ≥ 2 over a field k of characteristic ≠ 2. Let

(i_{0}(q) = 0, i_{1}(q), i_{2}(q), ⋯, i_{h(q)}(q) = [n/2])

denote the splitting pattern of q; that is, i_{0}(q) < i_{1}(q) < ⋯ < i_{h(q)}(q) are all distinct Witt indices that occur for q over arbitrary field extensions of k. We refer to M. Knebusch’s originating work on generic splitting of quadratic forms, [12], [13]. The index i_{j}(q) occurs over the field K_{j}, 0 ≤ j ≤ h(q), in a generic splitting tower K_{0} = k, K_{1} = k(q), ⋯, K_{h(q)} of q over k, where k(q) is the function field of q.

* Partially supported by LEQSF grant 075A. The first named author also is obliged to the SFB 343, University of Bielefeld, sponsored by the Deutsche Forschungsgemeinschaft.

In general, it is still a wide open problem to determine all tuples of strictly increasing integers that are splitting patterns of quadratic forms. However, for an excellent quadratic form q over k, the splitting pattern is known in arbitrary dimension, [12, Section 7], [8, Cor. 2.8]. It depends only on the dimension n of q; we denote it by

(i_{0}(n) = 0, i_{1}(n), ⋯, i_{h(n)}(n) = [n/2])

and have for the smallest positive Witt index i_{1}(n) :

1.1 Example. Let q be an anisotropic excellent quadratic form of dimension n over k. Write n = 2^{r} + d, where 2^{r} is the largest power of 2 less than n. Then the Witt index i_{1}(n) of q over its function field k(q) is given by

i_{1}(n) = d.

For arbitrary anisotropic forms we recall from [6], [9, Cor. 1.13]:

1.2 Theorem. Let q be an anisotropic quadratic form of dimension n over k. Then:

i_{1}(q) ≤ i_{1}(n).

In particular, we have by 1.1 and 1.2 for any field k:

1.3 Corollary. Let q be an anisotropic quadratic form of dimension n = 2^{r} + 1 over k. Then:

i_{1}(q) = 1.

It is natural to ask whether the inequality in 1.2 generalizes for higher Witt indices. Consider anisotropic quadratic forms q of dimension n with h(n) ≥ 2. In other words, n ≠ 2^{r}, 2^{r} − 1, and this implies h(q) ≥ 2 by the characterization of all forms q of height h(q) = 1 in [12, Thm. 5.8]. So, the indices i_{2}(n) and i_{2}(q) are defined, and we obtain:

1.4 Proposition. Let q be an anisotropic quadratic form over k of dimension n with h(n) ≥ 2. If i_{1}(q) = i_{1}(n), then

i_{2}(q) ≤ i_{2}(n).

Proof. Let q_{1} denote the anisotropic kernel of q over its function field k(q). We have

i_{2}(q) = i_{1}(q) + i_{1}(q_{1}).

On the other hand,

i_{2}(n) = i_{1}(n) + i_{1}(n − 2i_{1}(n))

since n−2i_{1}(n) is the dimension of the anisotropic kernel of an n-dimensional anisotropic excellent quadratic form over its function field.

By assumption, i_{1}(q) = i_{1}(n). Thus our claim i_{2}(q) ≤ i_{2}(n) amounts to showing i_{1}(q_{1}) ≤ i_{1}(n − 2i_{1}(n)). Again by the assumption, the last inequality is equivalent to

i_{1}(q_{1}) ≤ i_{1}(n − 2i_{1}(q));

that is,

i_{1}(q_{1}) ≤ i_{1}(dim q_{1}),

which is the result of Thm. 1.2 applied to the anisotropic from q_{1} over k(q).

By [9, Cor. 1.13], our assumption i_{1}(q) = i_{1}(n) in Prop.1.4 implies that q becomes an anisotropic Pfister neighbor over some field extension of k. The proof of the proposition carries over to the analogous result for higher Witt indices:

1.5 Corollary. Let q be an anisotropic quadratic form over k of dimension n with min {h(n), h(q)} ≥ j + 1. If i_{j}(q) = i_{j}(n), then

i_{j+1}(q) ≤ i_{j+1}(n).

Proof. Replace q_{1} in the proof of Prop.1.4 by q_{j}, the anisotropic kernel of q over the field K_{j} in a generic splitting tower of q over k. We then have

i_{j+1}(q) = i_{j}(q) + i_{1}(q_{j}),
i_{j+1}(n) = i_{j}(n) + i_{1}(n − 2i_{j}(n))

and application of Thm. 1.2 to q_{j} yields the claim.

Inspection of the list of splitting patterns in low dimensions, see 2.3 below with the patterns of excellent forms in each dimension given first, shows that the assumption i_{j}(q) = i_{j}(n) in Cor. 1.5 is not necessary for anisotropic forms q of dimension n ≤ 10.

1.6 Question. Can one drop in Cor. 1.5 the assumption that i_{j}(q) = i_{j}(n)? No counterexample exixts for forms q of dimension n ≤ 18.

2. Splitting patterns of trace forms.

Let E be a commutative étale algebra of dimension n over a field k of characteristic ≠ 2; so E is a product of separable extension fields of k. The trace form ⟨E⟩ of E, defined by

⟨E⟩(x) = Tr_{E/k}(x^{2}) for x ∈ E,

is a regular n-dimensional quadratic form over k; we just refer to [1], [2], [15].

So far, a basic question, which quadratic forms are (isometric to) trace forms, has been answered over arbitrary fields only in low dimensions. For a regular quadratic form q = ⟨a_{1}, ⋯, a_{n}⟩ over k let w_{m}(q), m = 0, 1, 2, ⋯, denote the Stiefel-Whitney invariants of q, [3], given in terms of cup products as

w_{m}(q) = ∑_{i_{1}<i_{2}<···<i_{m}}(a_{i_{1}}) ∪ ⋯ ∪ (a_{i_{m}}) ∈ H^{m}(k) = H^{m}(Gal (k_{s}/k), ℤ/2ℤ),

where k_{s} denotes a separable closure of k. The Galois cohomology groups H^{0}(k), H^{1}(k), and H^{2}(k) are isomorphic to ℤ/2ℤ, k^{∗}/k^{∗2}, and Br_{2}(k); thus w_{0}(q), w_{1}(q), and w_{2}(q) are the invariants given by the dimension mod 2, determinant d = d(q), and Hasse-Witt invariant of q, respectively. We summarize the characterization due to M. Epkenhans, M. Krüskemper, and J.-P. Serre, compare [1, Thm. 11]:

2.1 Theorem. A regular quadratic form q over k of dimension n ≤ 7 is (isometric to) a trace form (of some étale k-algebra E) if and only if, over k, a. n = 1 and q ≅ ⟨1⟩, b. n = 2 and q contains ⟨2⟩, c. n = 3 and q contains ⟨1, 2⟩, d. n = 4, q contains ⟨1⟩, and w_{3}(q) = 0, e. n = 5, q contains ⟨1, 1⟩, and w_{3}(q) = 0, f. n = 6, q contains ⟨1, 2⟩, and, over k(\sqrt{2d}), q contains ⟨1, 1, 2⟩, g. n = 7, q contains ⟨1, 1, 2⟩, and, over k(\sqrt{2d}), q contains ⟨1, 1, 1, 2⟩.

For n ≤ 5 the results can be found in [5] and [16]; the n = 6 characterization from [17] was afterwards independently discovered and proved in [4]; the result for n = 7 then follows from a construction in [14].

By considering orthogonal sums of trace forms, we immediately obtain from 2.1 for arbitrary dimensions:

2.2 Lemma. Let q be a regular quadratic form over k of dimension n. Let m = [n/2] and put

q_{0} = \{ \begin{matrix}m × ⟨1⟩ & if n ≡ 0 mod 4 \\ (m + 1) × ⟨1⟩ & if n ≡ 1 mod 4 \\ (m − 1) × ⟨1⟩ ⊥ ⟨2⟩ & if n ≡ 2 mod 4 \\ m × ⟨1⟩ ⊥ ⟨2⟩ & if n ≡ 3 mod 4. \end{matrix}

If q contains q_{0}, then q is a trace form.

Proof. By 2.1.d, every regular quadratic form q of dimension four that contains ⟨1, 1⟩ is a trace form. Thus every regular q of dimension n ≡ 0 mod 4 that contains m × ⟨1⟩ is a trace form. The proof is analogous for n ≡ 1, 2, or 3 mod 4 based on 2.1 a – d.

The following is a complete list of splitting patters in dimension at most ten.

2.3 Summary. The splitting patterns of anisotropic quadratic forms q of dimension n ≤ 10 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)
n = 9 (0,1,4), (0,1,2,3,4)
n = 10 (0,2,4,5), (0,2,3,5), (0,1,3,5), (0,2,3,4,5), (0,1,2,4,5), (0,1,2,3,5), (0,1,2,3,4,5).

Proof. By [11] or by [7], the tuples (0,2,3,4) and (0,1,5) fail to be splitting patterns in dimension eight and ten, respectively (see also [10]). Hence, for n ≤ 9, the list follows from [9, Ex.1.16]. For n = 10 the splitting pattern of excellent forms is (0,2,4,5), thus i_{1}(10) = 2 and we conclude from Thm. 1.2 that i_{1}(q) is 1 or 2 for every anisotropic quadratic form of dimension ten. There are at most four splitting patterns in dimension ten with i_{1}(q) = 1 since there are exactly four splitting patterns in dimension eight, the ones different from (0,4), that would not give rise to the excluded tuple (0,1,5).

There are at most three splitting patterns in dimension ten with i_{1}(q) = 2 since there are exactly three splitting patterns in dimension six. Thus, every splitting pattern in dimension ten is one of the seven tuples listed above. Moreover, all of the seven tuples occur as splitting patterns of quadratic forms in dimension 10; see Ex.3.1 below.

We have set up 2.1–2.3 in order to investigate which splitting patterns of quadratic forms are splitting patterns of trace forms. It might be a bit of a surprise to learn:

2.4 Theorem. All splitting patterns of (anisotropic) quadratic forms of dimension n ≤ 9 can be realized by trace forms.

Proof. In any dimension n, the sum n×⟨1⟩ of n squares is a trace form; it is anisotropic for example over ℚ. Hence we can realize in each dimension the splitting pattern of excellent forms, always listed first in 2.3, by a trace form over ℚ. In particular, this settles the dimensions n = 2, 3, and 5.

For n = 4 the splitting pattern (0,1,2) can be realized by the anisotropic quadratic form ⟨1, 1, 1, X⟩ over ℚ(X), say. By 2.2, this form is a trace form.

For n = 6 consider the quadratic forms ⟨1, 1, 2, X, Y, −2XY ⟩ and ⟨1, 1, 1, 1, X, Y ⟩ over k = ℚ(X, Y). Both of them are trace forms, by 2.2. The first one is isometric to ⟨1, 1, 2, X⟩ ⊥ Y ⟨1, −2X⟩, hence it is anisotropic over ℚ(X, Y) since ⟨1, 1, 2, X⟩ and ⟨1, −2X⟩ are anisotropic over ℚ(X). Clearly, the second one is anisotropic over ℚ(X, Y), too.

The form ⟨1, 1, 2, X, Y, −2XY ⟩ is of (signed) discriminant +1; that is, it is an Albert form, and Witt index 2 does not occur in its splitting pattern. So, by 2.3, its splitting pattern is (0,1,3).

The form ⟨1, 1, 1, 1, X, Y ⟩ splits exactly one hyperbolic plane over k(\sqrt{−XY}) and since it is of discriminant ≠ 1, Witt index 2 will occur, too. So its splitting pattern is (0,1,2,3).

For n = 7 the quadratic form ⟨1, 1, 1, 1, 1, X, Y ⟩ is a trace form, by 2.2. It is anisotropic over k = ℚ(X, Y) and splits exactly one hyperbolic plane over k(\sqrt{−XY}). Thus, by 2.3, its splitting pattern is (0, 1, 2, 3).

For n = 8 we have to realize the four splitting patterns in 2.3 that are not patterns of excellent forms. Consider the quadratic forms q_{1} = 6 × ⟨1⟩ ⊥ ⟨X, X⟩, q_{2} = 5 × ⟨1⟩ ⊥ ⟨X, Y, XY ⟩, q_{3} = 7 × ⟨1⟩ ⊥ ⟨X⟩, q_{4} = 6 × ⟨1⟩ ⊥ ⟨X, Y ⟩ over k = ℚ(X, Y). All of them are anisotropic over ℚ(X, Y); we check it for q_{2} : since ⟨1, 1, 1, 1, 1, X⟩ and ⟨1, X⟩ are anisotropic over ℚ(X), the form q_{2} = ⟨1, 1, 1, 1, 1, X⟩ ⊥ Y ⟨1, X⟩ is anisotropic over ℚ(X, Y).

The form q_{1} = 6 × ⟨1⟩ ⊥ ⟨X, X⟩ is isometric to ⟨1, 1, 1, X⟩ ⊗ ⟨1, 1⟩, thus q_{1} ≅ q′ ⊗ ⟨1, 1⟩ with dim q′ = 4. Moreover if, for some extension field K of k = ℚ(X, Y), the form q_{K} is isotropic we may choose q′_{K} to be isotropic. Thus q_{1} has splitting pattern (0,2,4), compare [9, Rem.3.3].

The form q_{2} = 5 × ⟨1⟩ ⊥ ⟨X, Y, XY ⟩ splits exactly one hyperbolic plane over K = ℚ(\sqrt{−7}). Namely, K has level 4 and, over K, we have 5 × ⟨1⟩ ≅ ⟨1, −1⟩ ⊥ ⟨−1, −1, −1⟩, so q_{2,K} ≅ H ⊥ ⟨−1, −1, −1, X, Y, XY ⟩ with a 6-dimensional anisotropic kernel ⟨−1, −1, −1, X, Y, XY ⟩. The form q_{2} splits exactly two hyperbolic planes over K = k(\sqrt{−XY}). Namely, over this extension field K of k, q_{2} is isometric to 4 × ⟨1⟩ ⊥ ⟨1, X, −X, −1⟩ ≅ H ⊥ H ⊥ 4 × ⟨1⟩ with a 4-dimensional anisotropic kernel 4 × ⟨1⟩. Since q_{2} has discriminant 1, over no extension field it will have Witt index 3, and hence q_{2} has splitting pattern (0,1,2,4).

The form q_{3} = 7 × ⟨1⟩ ⊥ ⟨X⟩ splits no or exactly one hyperbolic plane over every extension field K of k of level ≥ 8, since 7 × ⟨1⟩ stays anisotropic over such a field K. The form q_{3} splits three or four hyperbolic planes over every extension field K of k of level ≤ 4, since 7 × ⟨1⟩ splits completely over such a field K. The Witt index of q_{3,K} is 1 over K = k(\sqrt{−X}), and the Witt index of q_{3,K} is 3 over K = k(\sqrt{−7}). Thus q_{3} has splitting pattern (0, 1, 3, 4).

The form q_{4} = 6×⟨1⟩ ⊥ ⟨X, Y ⟩ splits exactly one hyperbolic plane over k(\sqrt{−XY}) with anisotropic kernel 6 × ⟨1⟩, it splits exactly two hyperbolic planes over k(\sqrt{−7}) with anisotropic kernel ⟨−1, −1, X, Y ⟩, and it splits exactly three hyperbolic planes over k(\sqrt{−1}) with anisotropic kernel ⟨X, Y ⟩. Hence q_{4} has splitting pattern (0,1,2,3,4).

This concludes the discussion of the 8-dimensional case.

For n = 9 it is only left, by 2.3, to realize the pattern (0,1,2,3,4) by a trace form. Consider the quadratic form 7 × ⟨1⟩ ⊥ ⟨X, Y ⟩ over ℚ(X, Y). By 2.2, this form is a trace form. It is anisotropic over ℚ(X, Y) and splits exactly two hyperbolic planes over ℚ(\sqrt{−X}, \sqrt{−Y}), hence Witt index 2 occurs in its pattern. Thus, by 2.3, its splitting pattern is (0,1,2,3,4).

The statement of Thm. 2.4 can be sharpened:

2.5 Corollary. All splitting patterns of (anisotropic) quadratic forms of dimension n ≤ 9 can be realized by trace forms of \underline{field} extensions of ℚ(X, Y).

Proof. The proof of Thm. 2.4 shows that for n ≤ 9 all splitting patterns can be realized by trace forms of some étale algebra over ℚ(X, Y). By the main result in [5], over Hilbertian fields K of characteristic zero, every trace form of an étale K-algebra of dimension n ≥ 3 is isometric to the trace form of some field extension of K. Thus it is only left to discuss the case n = 2. The trace form ⟨2, 2d⟩ of a real quadratic extension ℚ(\sqrt{d}) of ℚ remains anisotropic over ℚ(X, Y) and has splitting pattern (0,1).

3. Appendix. We illustrate how to realize all seven splitting patterns in dimension ten listed in 2.3. The following example can also be obtained from the classification of splitting patterns in [7].

3.1 Example. The form 10 × ⟨1⟩ over ℚ has pattern (0,2,4,5), the form

⟨1, X, Y, Z, T, XZ, XT, Y Z, Y T, XY ZT ⟩

over ℚ(\sqrt{−1})(X, Y, Z, T) has pattern (0,2,3,5), the form 7 × ⟨1⟩ ⊥ ⟨X, Y, −XY ⟩ over ℚ(X, Y) has pattern (0,1,3,5), the form 7 × ⟨1⟩ ⊥ ⟨X, X, X⟩ over ℚ(X) has pattern (0,2,3,4,5), the form 8 × ⟨1⟩ ⊥ ⟨X, Y ⟩ over ℚ(X, Y) has a pattern (0,1,2,4,5), the form 5 × ⟨1⟩ ⊥ ⟨2⟩ ⊥ ⟨X, Y, Z, −2XY Z⟩ over ℚ(X, Y, Z) has pattern (0,1,2,3,5), and the form 6 × ⟨1⟩ ⊥ ⟨X, Y, Z, T ⟩ over ℚ(X, Y, Z, T) has pattern (0,1,2,3,4,5).

It follows from 2.2 that, except for the second pattern (0,2,3,5), all patterns in 3.1 have been realized by trace forms. The form we have chosen to realize (0,2,3,5) is the neighbor of the Pfister form ⟨⟨X, Y, Z, T ⟩⟩ given by the complement of the Albert form ⟨XY, ZT, XY T, XY Z, XZT, Y ZT ⟩ over ℚ(\sqrt{−1})(X, Y, Z, T). We do not know if also (0,2,3,5) is the splitting pattern of some 10-dimensional trace form.

The splitting pattern of an excellent quadratic form is an example of a pattern that can be realized in arbitrary dimension n by a trace form, namely by n × ⟨1⟩ over ℚ, say. In this regard, we add:

3.2 Proposition. For every dimension n, the splitting pattern (0, 1, 2, 3, 4, ⋯, [n/2]) can be realized by a trace form.

Proof. Given n, let m = [n/2] and consider the quadratic form q_{0} over ℚ defined in 2.2. The dimension of q_{0} is n − m. Let φ be the quadratic form ⟨X_{1}, X_{2}, ⋯, X_{m}⟩ over ℚ(X_{1}, X_{2}, ⋯, X_{m}). Then

q := q_{0} ⊥ φ

is an anisotropic, n-dimensional quadratic form over k = ℚ(X_{1}, X_{2}, ⋯, X_{m}), and q is a trace form by 2.2. Over k(\sqrt{−X_{1}}), q has Witt index 1; over k(\sqrt{−X_{1}}, \sqrt{−X_{2}}), q has Witt index 2 (if n ≥ 4), and so on. Since q_{0} contains (m − 1) × ⟨1⟩, all consecutive integers 0, 1, 2, ⋯, m − 1, m = [n/2] occur in the splitting pattern of q.

In view of the statements 2.4, 3.2, one might want to ask:

3.3 Question. Are all splitting patterns of (anisotropic) quadratic forms realizable by trace forms?

References.

[1] E. Bayer-Fluckiger, Galois Cohomology and the Trace Form, Jber. Dt. Math.- Verein. 96 (1994), 35–55.

[2] P. E. Conner and R. Perlis, A Survey of Trace Forms of Algebraic Number Fields, World Scient. Publ. 2, Singapore, 1984.

[3] A. Delzant, Définition des Classes de Stiefel-Whitney d’un Module Quadratique sur un Corps Caractéristique Différente de 2, C. R. Acad. Sci. Paris 255 (1962), 1366–1368.

[4] M. Epkenhans, private communication, August 3, 1994.

[5] M. Epkenhans and M. Krüskemper, On Trace Forms of Étale Algebras and Field Extensions, Math. Z. 217 (1994), 421–433.

[6] D. W. Hoffmann, Isotropy of Quadratic Forms over the Function Field of a Quadric, Math. Z. 220 (1995), 461–466.

[7] D. W. Hoffmann, Splitting of Quadratic Forms, I: Forms of Small Dimension, preprint.

[8] J. Hurrelbrink and U. Rehmann, Splitting Patterns of Excellent Quadratic Forms, J. reine angew. Math. 444 (1993), 183–192.

[9] J. Hurrelbrink and U. Rehmann, Splitting Patterns of Quadratic Forms, Math. Nachr. 176 (1995), 111–127.

[10] J. Hurrelbrink and U. Rehmann, Quadratic forms of small dimension, preprint.

[11] O. Izhboldin, On the Isotropy of Quadratic Forms over the Function Field of a Quadric, preprint (1995).

[12] M. Knebusch, Generic Splitting of Quadratic Forms, I, Proc. London Math. Soc. 33 (1976), 65–93.

[13] M. Knebusch, Generic Splitting of Quadratic Forms, II, Proc. London Math. Soc. 34 (1977), 1–31.

[14] J.-F. Mestre, Extensions Régulières de ℚ(t) de Groupe de Galois Ã_{n}, J. Algebra 131 (1990), 483–495.

[15] J.-P. Serre, L’invariant de Witt de la Forme T r(x^{2}), Comment. Math. Helv. 59 (1984), 651–676.

Splitting Patterns and Trace Forms 8

[16] J.-P. Serre, Texts on Étale Algebras, 1991/92, unpublished.

[17] J.-P. Serre, Résumé des Cours au Collége de France, 1993/94, preprint.

Department of Mathematics  Fakultät für Mathematik
Louisiana State University  Universität Bielefeld
Baton Rouge, LA 70803  33501 Bielefeld
USA  Germany
jurgen@marais.math.lsu.edu  rehmann@mathematik.uni-bielefeld.de