THE WITT RING KERNEL FOR A

FOURTH DEGREE FIELD EXTENSION

A.S. Sivatski

Department of Mathematics St.Petersburg Electrotechnical University 197376, St.Petersburg, Russia

Abstract. We compute the Witt ring kernel for an arbitrary field extension of degree 4 and characteristic different from 2 in terms of the coefficients of a polynomial determining the extension. In the case where the lower field is not formally real we prove that the intersection of any power n of its fundamental ideal and the Witt ring kernel is generated by n-fold Pfister forms.

Let F be a field of characteristic different from 2. As usual denote by W (F) the Witt ring of F, i.e. the ring of equivalence classes of quadratic forms over F. In this note we compute the kernel W (L/F) of the restriction map W (F) → W (L), where L/F is an arbitrary field extension of degree 4. This kernel was computed in [2] in the case where L/F is a tower of two quadratic extensions, and our result can be considered as its generalization. The main step is to show that W (L/F) is generated as an ideal by 1-fold and 2-fold Pfister forms. The last statement is a particular case of a stronger result (see Corollarry 2), which as far as we know, has not been known even for biquadratic extensions.

Our notation is standard, and the main reference is the book [4]. Notice only that all the fields considered below are of characteristic different from 2. We write φ ≃ ψ if quadratic forms φ and ψ are isomorphic, and φ = ψ if their classes in the Witt ring are equal. The anisotropic part of a form φ is denoted by φ_{an}. We use the sign ⊥ for the direct sum of two forms and the signs + and − for the sum and the difference of elements of the Witt ring. For a form φ over a field F we denote by D_{F} (φ) the set of nonzero values of F represented by φ. By the n-fold Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩ we mean the product ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩. If V is a linear space over a field F and L/F is a field extension, then by definition V_{L} = L ⊗_{F} V.

We begin by the following statement, which plays the main role in the sequel.

Typeset by AMS-TEX

Key words and phrases. Witt ring, Pfister form, polynomial.

The work under this publication was partially supported by Royal society Joint Project”Quadratic forms and central simple algebras under field extensions” and Emory University.

Proposition 1. Suppose p(t) is an irreducible even degree polynomial over a field F, and θ is its root. Let k be a nonnegative integer, φ an anisotropic quadratic form over F such that dim φ ≥ \frac{k}{2} deg p + 1 and dim(φ_{F (θ)})_{an} = k. Then there exists a subform φ_{0} ⊂ φ and x ∈ F^{∗} such that dim φ_{0} ≤ \frac{1}{2} deg p + 1 and xp(t) ∈ D(φ_{0}_{F (t)}). In particular, the form φ_{0}_{F (θ)} is isotropic.

Proof. Let U be the linear space associated to φ, U = ⟨e_{1}, …, e_{dim φ}⟩. Let further V = ⟨v_{1}, …, v_{\frac{1}{2} (dim φ−k)}⟩ be a totally isotropic subspace of U_{F (θ)}. Put W =

∑_{i=0}^{\frac{1}{2} deg p} θ^{i}U. We can consider V and W as F - linear subspaces of the F - linear space

U_{F (θ)}. Obviously,

dim_{F} U_{F (θ)} = dim φ deg p, dim_{F} V = \frac{1}{2}(dim φ−k) deg p, dim_{F} W = (\frac{1}{2} deg p+1) dim φ.

Since dim φ > \frac{1}{2}k deg p, we have

dim_{F} V + dim_{F} W > dim_{F} U_{F (θ)}.

Hence V ∩ W ≠ 0. Let 0 ≠ v = ∑_{i=0}^{\frac{1}{2} deg p} θ^{i}u_{i} ∈ V, where u_{i} ∈ U. Denote by φ_{0}

the subform of φ determined by the F -linear subspace U_{0} = ⟨u_{0}, … u_{\frac{1}{2} deg p}⟩ ⊂ U.

Obviously, dim φ_{0} ≤ \frac{1}{2} deg p + 1. Put ṽ = ∑_{i=0}^{\frac{1}{2} deg p} t^{i}u_{i} ∈ F [t] ⊗_{F} U_{0}. Since φ(v) = 0,

we have p|φ_{0}(ṽ). Since deg φ_{0}(ṽ) ≤ deg p, we get that φ_{0}(ṽ) = xp for some x ∈ F. Finally, x ≠ 0, because v ≠ 0, and φ is anisotropic over F. □

We apply Proposition 1 to field extensions of degree 4.

Corollary 2. Under hypothesis of Proposition 1 suppose deg p = 4. Then φ = ψ + ∑ π_{i}, where dim ψ ≤ 2k, each π_{i} is similar to either a 2-fold or a 1-fold Pfister form and π_{i}_{F (θ)} = 0. Moreover, we can assume that each π_{i} is similar to a 2-fold Pfister form except the case where k = 0, the extension F (θ)/F contains a quadratic subextension F (\sqrt{d})/F such that φ_{F (\sqrt{d})} = 0, and disc(φ) = d. In this case we can assume that exactly one π_{i} is similar to a 1-fold Pfister form ( obviously, then disc(π_{i}) = d).

In particular, if k = 0, i.e. φ ∈ W (F (θ)/F), we have ψ = 0, and φ = ∑ π_{i}.

Proof. Assume first that k ≥ 1. By Proposition 1 we get φ ≃ φ_{0} ⊥ φ_{1}, where dim φ_{0} = 3 and φ_{0}_{F (θ)} is isotropic. Let π ≃ φ_{0} ⊥ det(φ_{0}). Then

φ = φ_{1} + ⟨− det(φ_{0})⟩ + π,

π is similar to a 2-fold Pfister form, π_{F (θ)} = 0, and

dim(φ_{1} + ⟨− det(φ_{0})⟩)_{an} ≤ dim φ − 2.

Therefore, we can go on by induction on dim φ. In the case k = 0 and dim φ ≥ 3 the same argument works. Finally, if k = 0 and dim φ = 2, then φ is similar to a 1-fold Pfister form. It is obvious that in this case the field F (θ) contains the field F (\sqrt{disc φ}). □

Remark. Let us call a form φ minimal with respect to an extension F (θ)/F, if dim ψ ≥ dim φ for any form ψ such that φ − ψ ∈ W (F (θ)/F). Corollary 2 claims that if deg p = 4, dim φ > 2k and dim(φ_{F (θ)})_{an} = k, then φ is not minimal. On the other hand, it is easy to see that if forms φ_{1} and φ_{2} over F are minimal with respect to the extension F (θ)/F, then the form φ_{1} ⊥ xφ_{2} over F ((x)) is minimal with respect to the extension F ((x))(θ)/F ((x)). Let τ be a 4-dimensional form over F such that the form τ_{F (θ)} is isotropic, but for any subform τ′ ⊂ τ of codimension 1 the form τ′_{F (θ)} is anisotropic. (The existence of such an extension F (θ)/F of degree 4 and a form τ has been established in [1] and [5]). In particular, τ is minimal. Therefore, the forms ψ_{1} ≃ τ ⊗ ⟨x_{1}, …, x_{m}⟩ and ψ_{2} ≃ τ ⊗ ⟨x_{1}, …, x_{m}⟩ ⊥ ⟨x_{m+1}⟩ over the field K = F ((x_{1}))((x_{2})) … ((x_{m+1})) are minimal, which shows that the inequality dim ψ ≤ 2k in Corollary 2 can not be improved. Therefore, in the case deg p = 4 the inequality dim φ ≥ \frac{k}{2} deg p + 1 in Proposition 1 can not be improved as well.

It remains to determine 2-fold Pfister forms in W (F (θ)/F). This can be done in terms of the coefficients of p.

Lemma 3. Let p(t) = t^{4} + at^{2} + bt + c be an irreducible polynomial over F, θ its root, π an anisotropic 2-fold Pfister form over F. Suppose that π_{F (\sqrt{d})} ≠ 0 for any d ∈ F^{∗} such that F (\sqrt{d}) ⊂ F (θ). Then the following two conditions are equivalent:

1) π_{F (θ)} = 0.

2) there exist x, y ∈ F^{∗} such that det ( \begin{matrix}−x & 0 & \frac{1}{2}(y − ax) \\ 0 & −y & −\frac{1}{2}bx \\ \frac{1}{2}(y − ax) & −\frac{1}{2}bx & −cx \end{matrix} ) ∈ F^{∗}^{2}

and π ≃ ⟨⟨x, y⟩⟩.

Proof. 1) ⟹ 2). Assume that π_{F (θ)} = 0. Let (V, π′) be the quadratic space of the pure subform of π, i.e. π ≃ ⟨1⟩ ⊥ π′. All 3-dimensional subforms of π are similar to each other. Therefore, by Proposition 1 there are x ∈ F^{∗} and v_{0}, v_{1}, v_{2} ∈ V such that

π′(t^{2}v_{2} + tv_{1} + v_{0}) = −xp.

We have φ_{F (θ)} ≠ 0 for any 2-dimensional subform φ ⊂ π′. Indeed, otherwise we would have F (\sqrt{disc φ}) ⊂ F (θ), hence π_{F (\sqrt{disc φ})} = 0, a contradiction to the hypothesis of the proposition. Therefore, the vectors v_{0}, v_{1}, v_{2} are linearly independent. Comparing the coefficients at the powers of t on both sides of the last equality we get

\{ \begin{matrix}π′(v_{2}, v_{2}) = −x, \\ π′(v_{1}, v_{2}) = 0, \\ π′(v_{1}, v_{1}) + 2π′(v_{0}, v_{2}) = −ax, \\ 2π′(v_{0}, v_{1}) = −bx, \\ π′(v_{0}, v_{0}) = −cx. \end{matrix}

Let π′(v_{0}, v_{2}) = \frac{1}{2}(y − ax). Then the matrix of the form π′ with respect to the

basis (v_{2}, v_{1}, v_{0}) is ( \begin{matrix}−x & 0 & \frac{1}{2}(y − ax) \\ 0 & −y & −\frac{1}{2}bx \\ \frac{1}{2}(y − ax) & −\frac{1}{2}bx & −cx \end{matrix} ). Since det π′ = 1, we get

π′ ≃ ⟨−x, −y, xy⟩, hence π ≃ ⟨⟨x, y⟩⟩.

2) ⟹ 1). We have π′(t^{2}v_{2} + tv_{1} + v_{0}) = −xp for some vectors v_{0}, v_{1}, v_{2} ∈ V. In particular, π′_{F (θ)} is isotropic, which means that π_{F (θ)} = 0. □

Corollary 4. Suppose a polynomial p(t) = t^{4} + at^{2} + bt + c is irreducible over a field F, and π is an anisotropic 2-fold Pfister form over F such that π_{F (θ)} = 0. Then at least one of the following two conditions hold:

1) π ≃ ⟨⟨d, e⟩⟩, where F (\sqrt{d}) ⊂ F (θ), e ∈ F^{∗}.

2) π ≃ ⟨⟨α(α − a)^{2}−4cα+b^{2}, −α⟩⟩ where α ∈ F, and (α(α − a)^{2}−4cα+b^{2})α ≠ 0.

Conversely, if π is a 2-fold Pfister form of one of two types above, then π_{F (θ)} = 0.

Proof. Assume that π_{F (θ)} = 0, and condition 1) does not hold. Then, as we have noticed already, the vectors v_{0}, v_{1}, v_{2} introduced in Lemma 3 are linearly independent. Put α = \frac{y}{x}. Since

det ( \begin{matrix}−x & 0 & \frac{1}{2}(y − ax) \\ 0 & −y & −\frac{1}{2}bx \\ \frac{1}{2}(y − ax) & −\frac{1}{2}bx & −cx \end{matrix} ) = x(\frac{1}{4}α(α − a)^{2} − αc + \frac{1}{4}b^{2}) ∈ F^{∗}^{2},

and ⟨⟨x, −x⟩⟩ = 0, we have

π ≃ ⟨⟨x, y⟩⟩ ≃ ⟨⟨x, αx⟩⟩ ≃ ⟨⟨x, −α⟩⟩ = ⟨⟨α(α − a)^{2} − 4cα + b^{2}, −α⟩⟩.

Conversely, if (α(α − a)^{2} − 4cα + b^{2})α ≠ 0, then put x = α(α − a)^{2} − 4cα + b^{2}, y = xα. The implication 2) ⟹ 1) of Lemma 3 shows that

⟨⟨α(α − a)^{2} − 4cα + b^{2}, −α⟩⟩_{F (θ)} = 0.
□

Remark. The polynomial f(α) = α(α − a)^{2} − 4cα + b^{2} is a cubic resolvent of the polynomial p.

Example. Suppose a ∈ F^{∗} \ F^{∗}^{2}, b, c ∈ F, b + 2c\sqrt{a} ∉ F (\sqrt{a})^{2}. Consider the quartic extension F (\sqrt{b + 2c\sqrt{a}})/F. If θ = \sqrt{b + 2c\sqrt{a}}, then

p(θ) = θ^{4} − 2bθ^{2} + (b^{2} − 4ac^{2}) = 0.

Let π be a 2-fold Pfister form such that π_{F (θ)} = 0, π_{F (\sqrt{a})} ≠ 0. By Corollary 4 we get that

π ≃ ⟨⟨α(α + 2b)^{2}−4(b^{2}−4ac^{2})α, −α⟩⟩ ≃ ⟨⟨(α + 2b)^{2}−4(b^{2}−4ac^{2}), −α⟩⟩ ≃ ⟨⟨ae^{2}+be+c^{2}, −e⟩⟩,

where e = \frac{4c^{2}}{α}. In fact we have got the result of theorem 3.9 from [1].

Proposition 5. Let L/F be a field extension of degree 4, and F is not formally real. Then for any n

W (L/F) ∩ I^{n}(F) = (W (L/F) ∩ I^{2}(F))I^{n−2}(F).

Hence in terminology of [1] the ideal W (L/F) ∩ I^{n}(F) is an n-Pfister ideal.

Proof. For any field K denote by G_{K} its absolute Galois group. Unfortunately the proof depends on a very deep result of Voevodsky, which claims that there exists an isomorphism I^{n}/I^{n+1}(K) ≃ H^{n}(G_{K}, ℤ/2ℤ), which takes an n-fold Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩ to the cup product (a_{1}) ∪ ⋯ ∪ (a_{n}) ([6], [7]). First we need a couple of lemmas.

Lemma 6. Let F be a field. The following two conditions are equivalent: 1) F is not formally real.

2) There exists a subfield F_{1} ⊂ F and a positive integer N such that I^{N}(F_{1}) = 0.

Proof. 2) ⟹ 1). If F is formally real, then so is any subfield F_{1} of F, hence I^{N}(F_{1}) ≠ 0 for any N.

1) ⟹ 2). Assume that F is not formally real, and s(F) = m = 2^{k} is its level, i.e. the minimal number m such that there exist x_{1}, …, x_{m} ∈ F with the equality x_{1}^{2} + ⋯ + x_{m}^{2} = −1. Let F_{0} be the prime subfield of F, i.e. F_{0} is either ℚ or ℤ/pℤ. Put F_{1} = F_{0}(x_{1}, …, x_{m}) ⊂ F, and N = m + k + 3. Let us show that I^{N}(F_{1}) = 0. Since cd_{2}(F_{0}(\sqrt{−1})) ≤ 2, we have cd_{2}(F_{1}(\sqrt{−1})) ≤ m + 2, hence I^{m+3}(F_{1}(\sqrt{−1})) = 0. Therefore, for any a_{1}, …, a_{m+3} ∈ F_{1}^{∗} we get

⟨⟨a_{1}, … a_{m+3}⟩⟩ ≃ ⟨⟨−1, b_{1}, …, b_{m+2}⟩⟩

for some b_{1}, … b_{m+2} ∈ F_{1}^{∗}. From this it follows at once that for any a_{1}, … a_{N} ∈ F_{1}^{∗}

⟨⟨a_{1}, … a_{N}⟩⟩ ≃ ⟨⟨−1⟩⟩^{⊗(k+1)} ⊗ ⟨⟨b_{1}, …, b_{N−k−1}⟩⟩

for some b_{1}, …, b_{N−k−1} ∈ F_{1}^{∗}. On the other hand, since x_{1}, … x_{m} ∈ F_{1}, we have ⟨⟨−1⟩⟩^{⊗(k+1)} = 0 ∈ W (F_{1}). This implies that ⟨⟨a_{1}, …, a_{N}⟩⟩ = 0 ∈ I^{N}(F_{1}), which finishes the proof. □

Lemma 7. The kernel of the restriction map I^{2}/I^{3}(F) → I^{2}/I^{3}(L) is generated by 2-fold Pfister forms belonging to W (L/F).

Proof. Assume that α ∈ ker(I^{2}/I^{3}(F) → I^{2}/I^{3}(L)) = ker(_{2}Br(F) →_{2} Br(L)). Obviously, ind α ≤ 4. Let φ be an Albert form associated to α. Then φ_{L} ∈ I^{3}(L). Since dim φ = 6, we conclude that φ ∈ W (L/F). In view of Corollary 2 the lemma is proved. □

We return to the proof of Proposition 5. It is proven in [3] (cor.18 and the remark after conjecture 20) that the kernel of the restriction map of the graded rings H^{∗}(G_{F} , ℤ/2ℤ) → H^{∗}(G_{L}, ℤ/2ℤ) is generated in degrees 1 and 2 as an ideal in H^{∗}(G_{F} , ℤ/2ℤ). Let α ∈ W (L/F)∩I^{n}(F). Assume that the conditions of Lemma 6 are fulfilled. Then it is easy to see that there is an intermediate subfield F_{1} ⊂ F̃ ⊂ F such that α ∈ W (LF̃/F̃) ∩ I^{n}(F̃) and cd_{2}(F̃) < ∞. Interpreting the graded ring H^{∗}(GF̃, ℤ/2ℤ) (resp. H^{∗}(G_{LF̃}, ℤ/2ℤ)) as the graded ring I^{∗}/I^{∗+1}(F̃) (resp. I^{∗}/I^{∗+1}(LF̃)) and using Lemma 7 we can prove Proposition 5 straightforwardly by decreasing induction on n.

Open question. Does Proposition 5 remain valid for a formally real field F ? Acknowledgements. This research has been carried out at Emory University,

Atlanta and the University of Nottingham in the framework of Royal society Joint Project ”Quadratic forms and central simple algebras under field extensions”. The author expresses his thanks to Detlev Hoffmann, Raman Parimala and Alexander Vishik for very fruitful discussions.

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[4] Scharlau W., Quadratic and Hermitian forms, Springer, Berlin Heidelberg New York (1985).

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e-mail: sivatsky@AS3476.spb.edu