Małgorzata Kruszelnicka

Uniwersytet Śląski

u-niezmiennik ciał formalnie rzeczywistych i nierzeczywistych

Praca semestralna nr 1

(semestr zimowy 2010/11)

Opiekun pracy: Tomasz Połacik

INSTYTUT MATEMATYKIUNIWERSYTET SLASKI

KATOWICE

Ma lgorzata Kruszelnicka

u-niezmiennik cia lformalnie rzeczywistych

i nierzeczywistych

Praca semestralnanapisana pod kierunkiemdr. hab. Tomasza Po lacika

Katowice 2010

INSTITUTE OF MATHEMATICSUNIVERSITY OF SILESIA

KATOWICE

Ma lgorzata Kruszelnicka

u-invariantof Formally Real

and Nonreal Fields

Semestral paperwritten by the supervision ofdr. hab. Tomasz Po lacik

Katowice 2010

u-invariant of Formally Real and Nonreal Fields

Ma lgorzata Kruszelnicka

Abstract

We introduce a notion o u-invariant of a field K. It is defined as amaximum of dimensions of all anisotropic quadratic forms over K. Forfields that are not formally real, we prove that u-invariant can not takevalues 3, 5, 7. As one of our main results, we prove Kneser’s Theoremfor nonreal fields. For formally real fields, we generalize the notion of u-invariant. Moreover, we extend Kneser’s Theorem to the case of arbitraryfields.

Keywords: u-invariant, quadratic forms, Witt ring, Pfister forms.

1 Introduction

One of the notions of algebraic theory of quadratic forms, which have been in-tensively studied in recent times is the notion of u-invariant of a field. Almostbefore our eyes the initial hypothesis that u-invariant is a power of 2 was over-thrown. Moreover, another hypothesis that u-invariant is an even number wasalso rejected.

Considering u-invariant of not formally real fields, Irving Kaplansky venturedin 1953 the conjecture that the u-invariant of a field is either infinity, or a powerof 2. This conjecture, known as Kaplansky’s Hypothesis, stood open for manyyears, until 1989, when Alexander S. Merkurjev disproved it by constructing afield of u-invariant 6.

Concerning parity of u-invariant, it was also proved that if F is a completediscretely valuated field with residue class field K of characteristic not 2, thenu(F ) = 2 · u(K), where u(K) denotes u-invariant of a field K. In particular,if K is algebraically closed and Fn = K((t1))((t2)) . . . ((tn)), then u(Fn) = 2n.This observation, due to I. Kaplansky, proved the existence of nonreal fields ofany 2-power u-invariant.

Still, as an open question the problem of existence of fields of u-invariantgreater than or equal to 9 remained, until 2000, when Oleg T. Izhboldin con-structed a field of u-invariant 9, proving, ipso facto, the existence of a field ofodd u-invariant greater than 1. These events led to my interest in the notion ofu-invariant, to which this paper is devoted.

The aim of the paper is to present the most important statements about u-invariant of formally real and nonreal fields. The paper is based on my MasterThesis, [3], defended in July 2009.

This paper is organized as follows. Section 2 contains necessary to under-stand the concept of the work notions that concern quadratic forms and basicproperties of quadratic forms useful in later discussions. Section 3 splits into

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two subsections. The first subsection presents a definition of the notion of u-invariant of a nonreal field. We proved there theorems that present relationshipbetween u-invariant of a field and the fundamental ideal of the Witt ring of afield. In particular, we present Kneser’s Theorem, which was first verified inspecial cases by I. Kaplansky, and, later, proved in full by Adolf Kneser. More-over, we prove also that u-invariant of a field can not take values 3, 5, 7. In thesecond subsection of Section 3 we discuss the notion of u-invariant of formallyreal fields. Namely, by appropriate theorems, we extend Kneser’s Theorem tothe case of arbitrary fields. Further generalizations of Kneser’s Theorem can befound in [2].

2 Preliminaries

The purpose of this chapter is to present basic notions and properties of quadra-tic forms. We shall confine to those facts that will be useful in subsequent con-siderations apart from the proof part. Throughout this paper we will consideronly fields of characteristic different from 2. The word field will be understoodas the field of characteristic different from 2.

2.1 Bilinear spaces and quadratic forms

Suppose that K is a fixed field. We define an (n-ary) quadratic form as thehomogeneous polynomial ϕ of degree 2 in n variables over K

ϕ(X1, . . . , Xn) =∑

1≤i<j≤ncijXiXj .

Taking aij = aji = 12cij for i, j ∈ {1, . . . , n}, i 6= j and aii = cii for i = 1, . . . , n

the quadratic form ϕ can be written in a symmetrical way

ϕ(X1, . . . , Xn) =n∑

i,j=1

aijXiXj .

By a matrix of coefficients of quadratic form ϕ we mean the matrix Mϕ = [aij ].If Mϕ is diagonal, the form ϕ is called a diagonal form. The form ϕ can bewritten also in the following way

ϕ(X1, . . . , Xn) = ϕ(X) = XTMϕX,

where X =

X1

...Xn

is a column of variables. When the matrix Mϕ is nonsingular, the form ϕis called a nonsingular form. We define a determinant of the form ϕ as thedeterminant of its matrix Mϕ and denote it by detϕ. Dimension of the form ϕwill be denoted by dimϕ.The n-dimensional forms ϕ,ψ will be called equivalent, denoted by ϕ ∼= ψ, whenthere exists an invertible matrix C ∈ Kn

n such that

ϕ(X) = ψ(CX), that is

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Mϕ = CTMψC.

It follows that if ϕ ∼= ψ, then detϕ = c2 detψ for some c 6= 0.There is a close relationship between quadratic forms and bilinear spaces

over K. We will present it below. But first let us recall that a bilinear spaceover the field K is defined as a pair (V, β), where V is a finite dimensionallinear space over K and β : V × V → K is a symmetric bilinear functional. If{v1, . . . , vn} is a basis of V , then the matrix A = [β(vi, vj)] is called the matrixof the functional β relative to the basis {v1, . . . , vn}.

Two bilinear spaces (V1, β1), (V2, β2) will be called isometric, if there existsan isomorphism f : V1 → V2 such that β1(u, v) = β2(f(u), f(v)) for every u, v ∈V1, and will be denoted by (V1, β1) ∼= (V2, β2) or V1

∼= V2.Suppose that ϕ ∈ K[X1, . . . , Xn] is an n-dimensional quadratic form over K

and V is an n-dimensional linear space over K with a fixed basis {v1, . . . , vn}.The form ϕ determines a functional Qϕ : V → K such that Qϕ(x1v1 + . . . +xnvn) = ϕ(x1, . . . , xn). It is easy to verify that the mapping βϕ : V × V → Kdefined as

βϕ(u,w) =12

(Qϕ(u+ w)−Qϕ(u)−Qϕ(w))

is a symmetric bilinear functional and the matrix Mϕ is the matrix of βϕ relativeto the basis {v1, . . . , vn}. Bilinear functional βϕ is nonsingular if the matrix Mϕ

is nonsingular. We define the bilinear space determined by the form ϕ as thepair (V, βϕ), while the pair (V,Qϕ) is called the quadratic space determined bythe form ϕ. By a quadratic functional determined by the form ϕ we mean thefunctional Qϕ.

It is easy to see that if we fix another space V ′ with the basis {w1, . . . , wn},or just another basis of the space V , then, received as above, bilinear space(V ′, β′ϕ) would be isometric with the space (V, βϕ). It is also easy to check thatif forms ϕ and ψ are equivalent, then they determine isometric bilinear spaces(V, βϕ), (V ′, βψ).

So, every n-dimensional quadratic form determines, uniquely up to isometry,an n-dimensional quadratic space (V,Qϕ). Therefore, in the later part thequadratic space (V,Qϕ) will be denoted by (V, ϕ).

It turns out that the reverse situation also occurs — every bilinear space(V, β) determines some quadratic form. Namely, suppose that (V, β) is a bilinearspace over K, that is V is a finite dimensional linear space and β : V × V → Kis a symmetric bilinear functional. The bilinear functional β determines themapping Qβ : V → K defined as Qβ(v) = β(v, v). The functional Qβ is calledthe quadratic functional determined by β.

Suppose that {v1, . . . , vn} is a fixed basis of V and A is the matrix of afunctional β in this basis. Then,

ϕ(X1, . . . , Xn) = [X1, . . . , Xn] ·A ·

X1

...Xn

determines a quadratic form over field K with the matrix of coefficients Mϕ = A,where βϕ = β. Notice that Qβ(x1v1 + . . .+xnvn) = ϕ(x1, . . . , xn). Moreover, if(V ′, β′) ∼= (V, β), then quadratic form ϕ′ determined by β′ would be equivalentto the form ϕ.

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In conclusion of these considerations, let us note that there exists one-onecorrespondence between quadratic forms and bilinear spaces, where equivalentforms correspond with isometric bilinear spaces.

This connection will be used in two ways in the sequel. First, having thequadratic form ϕ we will be using the geometric tools with reference to thequadratic space (V, ϕ) and the bilinear space (V, βϕ). And conversely, fromconsiderations about the bilinear space (V, β) we will conclude some facts aboutthe quadratic form ϕ determined by β.

Now suppose that ϕ is an n-dimensional quadratic form and V is an n-dimensional linear space with fixed basis {v1, . . . , vn}. Let (V, βϕ) be the bilinearspace determined by the form ϕ. As we know, every bilinear space V overthe field K has an orthogonal basis, which means that the form ϕ over K isequivalent to some diagonal quadratic form. The form ϕ of matrix equal todiag(a1, . . . , an) will be denoted by ϕ = 〈a1, . . . , an〉. That is 〈a1, . . . , an〉 =a1X

21 + . . . + anX

2n. This observation shows that, up to equivalent forms, we

can limit our considerations to diagonal forms.Suppose that ϕ,ψ are the n and m-dimensional quadratic forms over K,

respectively. Let Mϕ,Mψ be the matrices of coefficients of the forms ϕ and ψ.The orthogonal sum of forms ϕ and ψ is an n+m-dimensional quadratic formϕ⊥ψ with the matrix of coefficients equal to

Mϕ⊥ψ = Mϕ⊥Mψ =[Mϕ 00 Mψ

].

Note that 〈a1, . . . , an〉⊥〈b1, . . . , bm〉 = 〈a1, . . . , an, b1, . . . , bm〉.The tensor product of forms ϕ and ψ is an n ·m-dimensional quadratic form

ϕ ⊗ ψ with the matrix of coefficients equal to the Kronecker product of thematrices Mϕ and Mψ, that is

Mϕ⊗ψ = Mϕ ⊗Mψ.

It is easy to notice that

〈a1, . . . , an〉 ⊗ 〈b1, . . . , bm〉 = 〈a1b1, . . . , a1bm, a2b1, . . . , anbm〉.

The product of forms ϕ = 〈a1, . . . , an〉 and ψ = 〈b1, . . . , bm〉 will be denoted byϕ · ψ = 〈a1, . . . , an〉 · 〈b1, . . . , bm〉. Note also that det(ϕ⊥ψ) = detϕ · detψ anddet(ϕ⊗ ψ) = detϕdimψ · detψdimϕ.

The above constructions have natural interpretations in terms of bilinearspaces. Namely, suppose that (V1, βϕ), (V2, βψ) are the bilinear spaces deter-mined by the forms ϕ and ψ. Then, the orthogonal sum of the forms ϕ and ψdetermines the bilinear space (V, βϕ⊥ψ) such that V = V1 ⊕ V2 and βϕ⊥ψ is thebilinear functional that satisfies

βϕ⊥ψ(v1 ⊕ v2, w1 ⊕ w2) = βϕ(v1, w1) + βψ(v2, w2).

Whereas, the tensor product of forms ϕ and ψ determines the bilinear space(V, βϕ⊗ψ) such that V = V1 ⊗ V2 and βϕ⊗ ψ is the only bilinear functional onV that satisfies

βϕ⊗ψ(v1 ⊗ v2, w1 ⊗ w2) = βϕ(v1, w1) · βψ(v2, w2).

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Remark 2.1. Let K be an arbitrary field. By the symbol K we will denotethe set K \ {0}, which is a multiplicative group of the field K. By the symbolK2 we will denote the set {x2 : x ∈ K}, which is a subgroup of K.

By a group of square classes of the field K we mean the quotient ring K/K2

and its rank denote by q(K). The elements of K/K2 are denoted by aK2 ∈K/K2, and called the square classes of the element a.The field K will be called quadratically closed, when K = K2, that is whenq(K) = 1.

We shall also need the following theorem in later paragraphs of this paper.For a proof see [5].

Theorem 2.2 (Witt’s Cancellation Theorem). Let K be a field and letc, a1, . . . , an, b1, . . . , bm ∈ K, c 6= 0. Then

〈c, a1, . . . , an〉 ∼= 〈c, b1, . . . , bm〉 ⇒ 〈a1, . . . , an〉 ∼= 〈b1, . . . , bm〉Suppose that ϕ is a quadratic form over the field K. A form ψ is called a

subform of ϕ, when there exists a form σ such that ϕ ∼= ψ⊥σ. The form

ϕ(X1, . . . , Xn) =n∑

i,j=1

aijXiXj

will be called isotropic over K, when there exist the elements x1, . . . , xn ∈ K,not all equal to 0, such that ϕ(x1, . . . , xn) = 0. The bilinear space (V, βϕ)determined by the form ϕ will be called isotropic when there exists 0 6= v ∈ Vsuch that Qϕ(v) = βϕ(v, v) = 0. Then, the vector v will be called isotropic, andthe space (V, βϕ) a isotropic space.

The binary quadratic form ϕ will be called a hyperbolic form, when ϕ isequivalent to the form 〈1,−1〉. We define a hyperbolic plane as the bilinearspace determined by the hyperbolic form and denote it by H. The orthogonalsum of m hyperbolic planes H⊥ . . .⊥H︸ ︷︷ ︸

m

will be called a hyperbolic space and

denoted by mH. By the above notions we have the following characterizationof hyperbolic forms.

Theorem 2.3. Let ϕ be a binary nonsingular quadratic form over a field K.The following statements are equivalent:

(1) ϕ ∼= 〈1,−1〉;(2) ϕ is isotropic;(3) detϕ ∈ −K2.

Now, let V be an arbitrary bilinear space over a field K. Witt’s Decompo-sition Theorem states that V splits into an orthogonal sum

V = Vt⊥Vh⊥Van,where Vt is a totally isotropic subspace, Vh is a hyperbolic or zero subspace,Van is an anisotropic subspace, and the isometry types of Vg, Vh, Van are alluniquely determined. If (V, β) is a nonsingular bilinear space, then V = Vh⊥Van,where Vh is a hyperbolic (or zero) space and Van is an anisotropic space. So,it is easy to notice that (V, β) is isotropic iff V contains as a subspace thehyperbolic plane H. It is also easy to prove that if the form ϕ = 〈a1, . . . , an〉,for a1, . . . , an ∈ K, is isotropic, then ϕ splits into ϕ = 〈1,−1〉⊥〈b3, . . . , bn〉 forsome b3, . . . , bn ∈ K.

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2.2 The Witt ring of the field K

In this section we briefly recall the construction of the Witt ring of the field K,without going into details. These details can be found in [5].

Nonsingular symmetric bilinear spaces U and V over the field K are calledsimilar, denoted by U ∼ V , if there exist hyperbolic spaces mH and nH suchthat

U⊥mH ∼= V⊥nH.It turns out that so defined relation of similarity is an equivalence relation.

So, we can consider the equivalence classes of the relation ∼. By a similarityclass of some fixed space U we mean the class of all spaces similar to U anddenote it by [U ]. If U ∼= A in some basis of the space U , then instead of [U ] wewill write [A]. In particular, if A = (a1, . . . , an), then we write [a1, . . . , an].

Let W (K) be the set of all similarity classes of the nonsingular symmetricbilinear spaces over the field K. In W (K) can be naturally defined the oper-ations of addition and multiplication of classes. Namely, for any nonsingularsymmetric spaces U and V

[U ] + [V ] := [U⊥V ],

[U ] · [V ] := [U ⊗ V ].

So defined set W (K) with operations of addition and multiplication defined asabove is a commutative ring with unity. It is called the Witt ring of the field K.Let (V, β) be a nonsingular bilinear space over a field K. By Witt’s Decompo-sition Theorem we have

V ∼= mH⊥Van,where Van is an anisotropic space. Then, of course, [V ] = [Van]. Note also thatif Van and V ′an are both anisotropic, then Van ∼ V ′an iff Van ∼= V ′an. Therefore,every element of the Witt ring is represented, uniquely up to isometry, by exactlyone anisotropic form.

Remark 2.4. Let ϕ be a nonsingular quadratic form and let (V, βϕ) be abilinear space determined by ϕ. The Witt class [V ] of the space V will be alsodenoted by ϕ. Thus, the symbol ϕ = 〈a1, . . . , an〉, where a1, . . . , an ∈ K, willmean, depending on the context, either the quadratic form ϕ = a1X

21 + . . . +

anX2n, or the quadratic functional ϕ defined on the space V , with fixed basis

{v1, . . . , vn}, as follows ϕ(x1v1 + . . .+xnvn) = a1x21 + . . .+anx

2n, or the element

of the Witt ring W (K) represented by the bilinear space (V, βϕ).

Now consider the mapping e : W (K)→ Z/2Z defined as follows

e([U ]) = dim U + 2Z ={

2Z , if dim U ≡ 0 (mod 2)1 + 2Z , if dim U ≡ 1 (mod 2)

It is easy to check that e is a ring epimorphism. So defined epimorphism iscalled the dimension-index. Consider the kernel of the dimension-index. Notethat it consists of the Witt classes represented by even-dimensional spaces

ker e = {[U ] ∈W (K) : e([U ]) = 0} = {[U ] ∈W (K) : dim U ≡ 0 (mod 2)}.

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Moreover, ker e is the maximal ideal of the Witt ring additively generated bythe set

{〈1, a〉 : a ∈ K}.Namely, for an arbitrary fixed nonzero element 〈a1, . . . , an〉 ∈ W (K) we havethe following representation

〈a1, . . . , an〉 = 〈1, a1〉+ . . .+ 〈1, an〉 − n〈1〉.

If 〈a1, . . . , an〉 ∈ ker e, then n = 2m is an even number, and therefore

〈a1, . . . , an〉 = 〈1, a1〉+ . . .+ 〈1, an〉 −m〈1, 1〉.

The kernel of the epimorphism e is called a fundamental ideal of the Witt ring,and is denoted by I(K).

2.3 Fundamental ideal and Pfister forms

In this section we shall present the notion of n-fold Pfister form and focus onthe relationship between the fundamental ideal of the Witt ring and Pfisterforms. There will be presented these properties, which will be useful in laterconsiderations.

Among the symmetric quadratic forms the Pfister forms are especially im-portant. By a 1-fold Pfister form we mean a binary form 〈1, a〉, where a ∈ K.For n ≥ 2, by an n-fold Pfister form we mean the tensor product of n 1-foldPfister forms

〈1, a1〉 ⊗ . . .⊗ 〈1, an〉, a1, . . . , an ∈ K,and denote it by

〈〈a1, . . . , an〉〉 := 〈1, a1〉 ⊗ . . .⊗ 〈1, an〉.

Note that if ϕ = 〈〈a1, . . . , an〉〉 is the n-fold Pfister form, then dimϕ = 2n, andif n ≥ 2, then detϕ = K2.

Let ϕ be a Pfister form over a field K. It is easy to show that if the form〈1,−1〉 is a subform of ϕ, then ϕ is hyperbolic. Generally, for an arbitraryanisotropic form ψ we have

〈1,−1〉 ⊗ ψ = ψ⊥− ψ ∼ 0.

To prove that property, it suffices to show that if (V, βψ) is a nonsingularsymmetric bilinear space determined by the form ψ, then (V, βψ)⊥(V,−βψ) is ahyperbolic space. Moreover, we have the following theorem (for a proof see [3]).

Theorem 2.5. Let ϕ be an arbitrary Pfister form over a field K. Then ϕ isisotropic if and only if it is hyperbolic.

Consider now the fundamental ideal I(K) of the Witt ring of the field K. Aswe noticed in an earlier paragraph, I(K) is additively generated by similarityclasses of all 1-fold Pfister forms. Let us recall that by a product of two ideals Iand J of the ring P we mean the ideal I · J generated by the set of all productsx · y, where x ∈ I, y ∈ J . In the case when I = J the product of ideals I · Jtakes the following form

I2 = {∑

xy : x, y ∈ I},

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where all the sums∑xy contain a finite number of components.

So, let 〈1, x〉, 〈1, y〉 ∈ I(K). The product of these classes 〈1, x〉〈1, y〉 =〈1, x, y, xy〉 lies in I2(K). Since the classes 〈1, x〉, 〈1, y〉 are arbitrary, so thesquare of fundamental ideal I2(K) is additively generated by the set

{〈1, x〉〈1, y〉 : a, b ∈ K}.

Looking at higher powers of the fundamental ideal, we can show that for anyn ≥ 2, In(K) is additively generated by similarity classes of all n-fold Pfisterforms. The proof of the following proposition can be found in [3].

Proposition 2.6. For any n ∈ N the ideal In(K) is additively generated by thesimilarity classes of all n-fold Pfister forms.

Analyzing forms from In(K) we will reach Arason–Pfister Theorem knownas Hauptsatz.

Theorem 2.7 (Arason–Pfister Hauptsatz). Let ϕ be a nonzero anisotropic formover a field K. If ϕ ∈ In(K), then dimϕ ≥ 2n.

Corollary 2.8. Let ϕ be a 2n-dimensional quadratic form over a field K. Thenϕ ∈ In(K) iff there exists an n-fold Pfister form ψ and a ∈ K such that

ϕ ∼= a · ψ

Proofs of the above theorem and corollary are beyond the scope of this work,so we omit them. They can be found in [4].

2.4 The set of nonzero values

In this paragraph we shall define a set of nonzero values of a quadratic formand present its properties. All of the proofs can be found in [3] or [5].

Let ϕ ∈ K[X1, . . . , Xn] be an n-dimensional quadratic form over a field K.The element a ∈ K is called the element represented by the form ϕ, when thereexist x1, . . . , xn ∈ K such that ϕ(x1, . . . , xn) = a.

The set of nonzero values of the form ϕ will be denoted by DK(ϕ). Di-rectly from this definition results that if the forms ϕ and ψ are equivalent, thenDK(ϕ) = DK(ψ).

Let (V, βϕ) be the bilinear space determined by the form ϕ. Then

DK(ϕ) = {βϕ(v, v) : v ∈ V } \ {0}

The setDK(ϕ) will be also denoted byDK(V ). In particular, if ϕ = 〈a1, . . . , an〉,then

DK(ϕ) = {a1x21 + . . .+ anx

2n : x1, . . . , xn ∈ K} \ {0}.

In this situation the set DK(ϕ) = DK(〈a1, . . . , an〉) will be denoted byDK(a1, . . . , an). The form ϕ will be called a universal form, if DK(ϕ) = K.

Let us take the element a ∈ DK(ϕ). Then there exists v ∈ V such thatQϕ(v) = a. For an arbitrary c ∈ K we have ac2 = Qϕ(cv) ∈ DK(ϕ). Therefore,if a ∈ DK(ϕ), then aK2 ⊂ DK(ϕ). Thus DK(ϕ) is a sum of some square classesof K. Moreover, the set DK(1, a) is a subgroup of the group K for any a ∈ K.In the case of hyperbolic form the set DK(1,−1) = K. Thus, any hyperbolicform is universal. If the form ϕ ∼= ψ⊥σ, then DK(ψ) ⊂ DK(ϕ).

We shall now quote some properties of the set of nonzero values.

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Theorem 2.9 (Representation Criterion). Let ϕ be a quadratic form over afield K and let a ∈ K. If a ∈ DK(ϕ), then there exist elements a2, . . . , an ∈ Ksuch that ϕ ∼= 〈a, a2, . . . , an〉.Proposition 2.10. Let K be an arbitrary field and let a, b, c ∈ K. Then,

(i) c ∈ DK(a, b) ⇐⇒ (a, b) ∼= (c, abc);

(ii) 1 ∈ DK(a, b) ⇐⇒ (a, b) ∼= (1, ab);

(iii) If a+ b 6= 0, then (a, b) ∼= (a+ b, ab(a+ b)).

Proof of the above proposition can be found in [3]. Now, let ϕ ∼= (a1, . . . , an),for a1, . . . , an ∈ K. As we know DK(ϕ) ⊆ K. Let a ∈ K and suppose thataϕ ∼= ϕ. Hence, for an arbitrary element c ∈ K such that a = c

a1we have:

ϕ ∼= aϕ ∼= (c, aa2, . . . , aan).

And therefore, c ∈ DK(ϕ). In this way, the following proposition has beenproven.

Proposition 2.11. If ϕ is a quadratic form over a field K such that aϕ ∼= ϕfor every a ∈ K. Then ϕ is a universal form.

We shall need also these two following facts (for proofs see [3]).

Proposition 2.12. Any binary form over a field K is universal iff any 1-foldPfister form over K is universal.

Proposition 2.13. If 1 ∈ DK(a, b) for every a, b ∈ K, then K = DK(a, b) forevery a, b ∈ K.

By the above statements, every nonsingular isotropic form over K is univer-sal (see [3]). Combining the condition of representation any element of the fieldK with quadratic forms we get the following theorem.

Theorem 2.14. Let ϕ be a quadratic form over a field K and let d ∈ K. Thend ∈ DK(ϕ) if and only if a form ϕ⊥〈−d〉 is an isotropic form.

Hence, by the above theorem, it results the following corollary, which com-bines the notions of isotropic and universal quadratic forms.

Corollary 2.15. For any n ∈ N the following statements are equivalent:

(i) Every n-dimensional quadratic form over K is a universal form;

(ii) Every n+ 1-dimensional quadratic form over K is an isotropic form.

Now, let ϕ be an arbitrary symmetric form. For a ∈ K we will consider theform aϕ. It is easy to notice, by the definition of the set of nonzero values, thatDK(aϕ) = a ·DK(ϕ). We will be most interested in the case when ϕ ∼= aϕ. Wedefine the set

GK(ϕ) = {a ∈ K : ϕ ∼= aϕ}as a set of similarity factors of the form ϕ. It turns out that the set GK(ϕ) is asubgroup of the group K and K2 ⊆ GK(ϕ). Moreover, GK(ϕ) ⊆ DK(ϕ) if andonly if 1 ∈ DK(ϕ). In the case when the form ϕ is a Pfister form, sets DK(ϕ)and GK(ϕ) are equal.

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Theorem 2.16. Let ϕ be an arbitrary Pfister form over a field K. Then

DK(ϕ) = GK(ϕ)

The proof can be found in [5].

3 u-invariant of nonreal and formally real fields

In this section we will present the definition of u-invariant of a field, its essentialfeatures and the connection between the notions of u-invariant and fundamentalideal I(K) of a Witt ring.

First, let us recall that by a formally real field we mean a field such that−1 ∈ K is not a sum of squares of elements from K. Otherwise, if −1 is a sumof squares of elements from K, then K is called a nonreal field.

3.1 u-invariant of a nonreal field

Let K be a nonreal field, that is −1 is a sum of squares of elements from K.So, there exist elements a1, . . . , an ∈ K such that

−1 = a21 + . . .+ a2

n.

The smallest number n ∈ N with this property is called a Pfister index of thefield K and it is denoted by s(K).

Let M be the set of dimensions of all anisotropic quadratic forms over K.

M = {dimϕ : ϕ is an anisotropic form over K}.

If M is bounded from above, we define u-invariant of a field K as a maximum ofM and denote it by u(K). If such a maximum does not exist, we put u(K) =∞.That is,

u(K) = max {dimϕ : ϕ is an anisotropic form over K},

if the maximum exists, and u(K) =∞ otherwise.

Example 3.1. Consider K = C. The form ϕ = 〈1〉 is anisotropic over C, whileevery form of dimension more than or equal to 2 is isotropic. Thus u(C) = 1.

Due to connection between universal and isotropic forms, we quote the fol-lowing proposition, where as the minimum of empty set we take ∞.

Proposition 3.2. Let K be a nonreal field.Then(1) u(K) = min {n ∈ N : every form ϕ of dimension > n is isotropic}(2) u(K) = min {n ∈ N : every form ϕ of dimension ≥ n is universal}

Proof. Results from corollary 2.15.

Having defined the notion of u-invariant let us characterise its features.What values can the u-invariant of a field K take? Let’s start our consider-ations with the case when u(K) = 1.

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Theorem 3.3. u(K) = 1 iff K is quadratically closed.

Proof. (⇐) Suppose that K is quadratically closed, that is K = K2. Let a ∈ K.Then DK(a) = aK2 = aK = K. Thus any quadratic form of dimension morethan or equal to 1 is universal, which shows that u(K) = 1.

(⇒) By the assumption that form 〈1〉 is universal, we have DK(1) = K. Onthe other hand DK(1) = K2, so K = K2.

Considering the possible successive values of u-invariant we will show thatu(K) /∈ {3, 5, 7} for any field K. But first let us present statements expressingthe relationship between u-invariant and the fundamental ideal of the Witt ringof K.

Theorem 3.4. Let K be an arbitrary field and let I(K) be the fundamentalideal of the Witt ring of K. Then I2(K) = 0 iff every binary form over K isuniversal.

Proof. (⇒) Suppose I2(K) = 0. For any a, b ∈ K we have 〈1,−a,−b, ab〉 ∈I2(K). Therefore 〈1,−a,−b, ab〉 = 0. Hence

(1,−a,−b, ab) ∼= (1,−1, 1,−1) ∼= (a,−a, b,−b)

By Witt’s Cancellation Theorem, we get (1, ab) ∼= (a, b). Thus, from Proposition2.10, we conclude that 1 ∈ DK(a, b). Referring to the Corollary 2.13 we getDK(a, b) = K. Hence every binary quadratic form over K is universal.

(⇐) From Corollary 2.6 it follows that I2(K) is additively generated bythe 2-fold products 〈1, a〉 ⊗ 〈1, b〉. It suffices to show that 〈1, a〉 ⊗ 〈1, b〉 = 0for every a, b ∈ K. By the assumption we have DK(a, b) = K. In particular−1 ∈ DK(a, b). Then by Theorem 2.10 (a, b) ∼= (−1,−ab). So, we obtain thefollowing class equality 〈a, b〉 = 〈−1,−ab〉. Hence we get

〈1, a〉〈1, b〉 = 〈1, a, b, ab〉 = 〈a, b〉+ 〈1, ab〉 = 〈−1,−ab〉+ 〈1, ab〉 = 0.

So, since a, b ∈ K are arbitrary, we get I2(K) = 0.

Theorem 3.5. Let K be an arbitrary field. Then

u(K) ≤ 2 ⇐⇒ I2(K) = 0.

Proof. Follows from the previous theorem.

On the basis of Theorems 3.4 and 3.5, we have the following conclusion.

Corollary 3.6. Let K be an arbitrary nonreal field. Then the following condi-tions are equivalent:

(i) u(K) ≤ 2;

(ii) Every binary form over K is universal;

(iii) Every 2-fold Pfister form over K is a hyperbolic form;

(iv) I2(K) = 0.

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From this corollary, we know that if the u-invariant of a field K is less thanor equal to 2, then I2(K) = 0. It turns out that this case can be generalized,however the proof of this generalization requires a strong statement – Hauptsatz(Theorem 2.7).

Theorem 3.7. Let K be an arbitrary field. If u(K) < 2n, then In(K) = 0.Moreover, if u(K) ≤ 2n, then every nonzero anisotropic form in In(K) is auniversal n-fold Pfister form.

Proof. First, suppose that u(K) < 2n. Let ϕ be any n-fold Pfister form over K.Since dimϕ = 2n > u(K), by the definition of u-invariant, ϕ is an isotropicform. By Theorem 2.5, ϕ is also a hyperbolic form. Since In(K) is additivelygenerated by classes of n-fold Pfister forms and the form ϕ is arbitrary, so wehave In(K) = 0.

Now suppose that u(K) ≤ 2n. Consider any nonzero anisotropic form ψ ∈In(K). Then dimψ ≤ 2n. From Theorem 2.7 it follows that dimψ ≥ 2n.Summarizing: dimψ = 2n ≥ u(K), which implies that ψ is a universal form.

Now we have to show that ψ is an n-fold Pfister form. By Corollary 2.8, theform ψ can be presented as follows

ψ ∼= a · σ,

where a ∈ K and σ is an n-fold Pfister form. Since dimσ = 2n ≥ u(K), theform σ is a universal form. Thus a ∈ DK(σ) = GK(σ). So, we obtain thefollowing equalities

ψ ∼= a · σ ∼= σ.

Therefore ψ is an n-fold Pfister form.

The next theorem concerns the notion of fundamental ideal I(K) and theissue of parity of u-invariant.

Theorem 3.8. Let K be an arbitrary field. If I3(K) = 0 and 1 < u(K) < ∞,then u-invariant of K is even.

Proof. Assume I3(K) = 0 and 1 < u(K) < ∞. Let σ ∈ I2(K). First we willshow that the form σ is universal. Indeed, for σ ∈ I2(K) i a ∈ K we have

〈1,−a〉 · σ ∈ I3(K) = 0.

From the above equality we get σ ∼= a · σ. Proposition 2.11 shows, in fact, thatσ is a universal form.

Suppose, reasoning by contradiction, that u = u(K) is odd. Let ϕ be a u-dimensional anisotropic form and a = detϕ. Since, by the definition of u-invariant, ϕ is a universal form, and we have:

ϕ ∼= 〈a〉⊥σ1∼= 〈−a〉⊥σ2,

for suitable (u−1)-dimensional forms σ1, σ2. Dimensions of σ1 and σ2 are u−1,so these forms are even-dimensional, and hence σ1, σ2 ∈ I(K). Furthermore

a = detϕ = a · detσ1 = −a · detσ2.

Hence detσ1 = 1 and detσ2 = −1. But, since dimσ1 = u − 1 = dimσ2, theneither σ1 is in I2(K), or σ2 is in I2(K). Assume that σ1 ∈ I2(K). By the

12

first part of the proof it follows that σ1 is universal. In that case the form σ1

presents the element −a. From the Representation Criterion σ1∼= 〈−a〉⊥ρ, for

some form ρ. So,ϕ = 〈a〉⊥〈−a〉⊥ρ,

which implies that ϕ is an isotropic form, a contradiction.

On the basis of statements presented so far we can now prove that u(K) /∈{3, 5, 7}.Theorem 3.9. Let K be an arbitrary field. Then u(K) /∈ {3, 5, 7}.Proof. Note that we don’t have to consider each of the above three cases sep-arately. Theorem 3.7 shows that if u(K) < 8, then I3(K) = 0. And from theprevious theorem u(K) is even, which leads to contradiction.

Remark 3.10. Considered in this paragraph fields are nonreal. So, let s(K) =n be a Pfister index of K. That means there is a presentation

−1 = a21 + . . .+ a2

n

for some a1, . . . , an ∈ K, where n ∈ N is the smallest number with this property.Hence, and by the definition of u-invariant, we get the following inequality

s(K) ≤ u(K).

Now we will present the connection between u-invariant of K and the cardi-nality of the group of square classes of K. But first, let us prove the followingauxiliary lemma.

Lemma 3.11 (Kneser). Let K be a nonreal field and let ϕ be an anisotropicd-dimensional quadratic form over K such that DK(ϕ) 6= K. Then for anya ∈ K

DK(ϕ) ( DK(ϕ⊥〈a〉).Moreover, ϕ represents at least d distinct square classes of K.

Proof. Let a ∈ K. Suppose that DK(ϕ) = DK(ϕ⊥〈a〉). Hence a ∈ DK(ϕ).Suppose −1 = e21 + . . .+ e2s, where s = s(K). Inductively on i we will show thata(e21 + . . .+ e2i ) ∈ DK(ϕ) for every i ∈ {1, . . . , n}.

(1) Since by the assumption a ∈ DK(ϕ), so for i = 1 we have ae21 ∈ DK(ϕ).(2) Suppose that i ≥ 2 and a(e21 + . . .+ e2i−1) ∈ DK(ϕ). Then

a(e21 + . . .+ e2i−1) + ae2i ∈ DK(ϕ⊥〈a〉) = DK(ϕ).

So, we showed that a(e21 + . . .+ e2i ) ∈ DK(ϕ) for every i ∈ {1, . . . , s}, therefore−a = a(e21 + . . . + e2s) ∈ DK(ϕ). By Theorem 2.14 we conclude that the formϕ⊥〈a〉 is isotropic, and therefore also universal. Hence

DK(ϕ) = DK(ϕ⊥〈a〉) = K,

a contradiction.Now, by induction on d, we will show that the form ϕ represents at least ddistinct square classes. Let ϕ = (a1, . . . , ad).

(i) For d = 1 the form ϕ = (a1) represents the class a1K2.

13

(ii) Suppose that d ≥ 2 and the form ϕ′ = (a1, . . . , ad−1) represents at leastd− 1 distinct square classes. Then

ϕ = (a1, . . . , ad−1, ad) = ϕ′⊥〈ad〉

The already proven part shows that DK(ϕ′) ( DK(ϕ′⊥〈ad〉), therefore ad /∈DK(ϕ′).

Now, let ϕ be a quadratic form over an arbitrary field K. Let us considerthe set of nonzero values of ϕ. If DK(ϕ) is a sum of m square classes, then thenumber m is denoted by V (ϕ). If m =∞, we put V (ϕ) =∞. That is,

V (ϕ) = card {aK2 : a ∈ DK(ϕ)}.

Notice that if the form ϕ is universal, that is DK(ϕ) = K, then V (ϕ) = q(K).Therefore, by Kneser’s Lemma 3.11, for an arbitrary anisotropic form ϕ we havedimϕ ≤ V (ϕ).

Theorem 3.12 (Kneser). For any nonreal field K we have the following in-equality

u(K) ≤ q(K).

Proof. Proof of Kneser’s Theorem follows directly from Kneser’s Lemma 3.11and the definition of u-invariant. Namely, for an arbitrary anisotropic form ϕwe have the following inequality:

dimϕ ≤ V (ϕ).

So, if ϕ ranges over all anisotropic forms over K, then:

max (dimϕ) ≤ q(K).

And therefore,u(K) ≤ q(K),

as desired.

Kneser’s Theorem and the earlier remark leads us to relationships betweenu(K), s(K) and q(K). Namely, there are such inequalities

s(K) ≤ u(K) ≤ q(K),

from which we can draw the following corollary.

Corollary 3.13. Witt ring W (K) of K is finite if and only if s(K) and q(K)are both finite.

Proof. (⇒) Assume W (K) is finite. Let a, b ∈ K and aK2 6= bK2. Note thatquadratic forms (a) and (b) are not isometric, and therefore, because they haveequal dimensions, present different elements of the Witt ring 〈a〉 6= 〈b〉.

Suppose that the group K/K2 is infinite. Then the Witt ring W (K) containsan infinite subset {〈a〉 ∈W (K) : a ∈ K}, which gives a contradiction.

14

Now suppose that the Pfister index s(K) of K is infinite. Then, for every n ∈N, the element −1 /∈ DK(n(1)). Thus, the form n(1) is anisotropic. Moreover,if n < m, then n(1) � m(1). Indeed, if n(1) ∼= m(1), then the form (m− n)(1)would be isotropic, which, as we have already noticed, is imposible. So W (K)contains an infinite set of elements {n〈1〉 : n ∈ N}, a contradiction.

(⇐) Let {a1K2, . . . , aqK2} be all of the distinct square classes. From theprevious theorem u = u(K) ≤ q. Thus every anisotropic form over K hasdimension less than or equal to q.

Notice that for k ∈ N, every k-dimensional form, uniquely up to isometry,appears as (ai1 , . . . , aik), where {i1, . . . , ik} ⊂ {1, . . . , q}. Thus,uniquely up toisometry, there are at most

(q+k−1k

)k-dimensional forms over K.

On the other hand, every nonzero element of the Witt ring is uniquely de-termined by the isomerty class of an anisotropic form. Therefore, every elementof W (K) is uniquely determined by some form of dimension less than or equalto u. Finally, the ring W (K) contains at most

1 +n∑

k=1

(q + k − 1

k

)

elements.

Theorem 3.14. Let K be a nonreal field such that q(K) <∞. Then

|W (K)| ≥ u(K) · q(K)

Proof. Assume that q(K) < ∞. By Theorem 3.12, we have u(K) < ∞. Ifs(K) = ∞, then W (K) is an infinite ring and the thesis is obvious. Supposethat s(K) <∞. Then W (K) is finite.

Let u = u(K) and ϕ be an anisotropic form over K of dimension dimϕ = u.Assume that ψ is a subform of ϕ of odd dimension. Then, of course, ψ is ananisotropic form over K. Moreover, for every a ∈ K we have

det(〈a〉ψ) = adimψ · detψ = adetψK2.

Thus, if aK2 6= bK2, then det(〈a〉ψ)K2 6= det(〈b〉ψ)K2. In this case, forms〈a〉ψ and 〈b〉ψ are not isometric, and because they have equal dimensions and areanisotropic, they represent different elements in W (K). So, for each odd numberk ≤ u there are at least q(K) elements in W (K) represented by anisotropic k-dimensional forms. Suppose that there are m odd numbers less than or equalto u. Then, there are at least m · q(K) different elements in W (K) representedby odd-dimensional forms.

On the other hand, we know that W (K)/I(K) ∼= Z2. Therefore, the numberof elements of W (K) represented by even-dimensional forms is equal to thenumber of elements represented by odd-dimensional forms, and consequently

|W (K)| ≥ 2 ·m · q(K).

If u is even, then m = u2 , and hence

|W (K)| ≥ u · q(K).

If u is odd, then m = 12 (u+ 1), and hence

|W (K)| ≥ (u+ 1) · q(K).

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3.2 General u-invariant

Suppose that K is a formally real field. Then −1 is not the sum of squares ofelements of K. So, for every m ∈ Z, the form m · 〈1〉 is anisotropic. And hence,u(K) =∞. Therefore, the aim of this paragraph is to generalize the definitionof u-invariant over the formally real fields.

First, let us present necessary definitions. Let W (K) be a Witt ring of afield K. Consider an arbitrary form ϕ ∈W (K). The form ϕ is called a torsionform, if the following condition is fulfilled:

∃n∈N nϕ = 0 ∈W (K).

It is easy to notice that if ϕ and ψ are torsion forms, then forms ϕ⊥ψ and ϕ⊗ψare torsion as well. Then, a subgroup Wt(K) of the Witt ring W (K) equal to

Wt(K) = {ϕ ∈W (K) : ∃n ∈ N nϕ = 0}

is called a torsion subgroup of additive group of the Witt ring W (K).Let K be an arbitrary field. Let M be the set of dimensions of all anisotropic

torsion forms over K,

M = {dimϕ : ϕ is an anisotropic form, ϕ ∈Wt(K)}.

If M is bounded from above, we define general u-invariant of a field K as amaximum of M and denote it by ug(K). If such a maximum does not exist, weput ug(K) =∞. That is

ug(K) = max{dimϕ : ϕ is an anisotropic form, ϕ ∈Wt(K)},

if the maximum exists, and ug(K) =∞ otherwise.Note that if K is the nonreal field, the above definition of u-invariant coin-

cides with the definition introduced in the earlier paragraph. This follows fromthe fact that any quadratic form over the nonreal field is a torsion form, andthen Wt(K) = W (K) and ug(K) = u(K). While, if K is the formally real field,then Wt(K) ⊆ I(K). So, any torsion form is even-dimensional. By the aboveconsiderations, in the later part of this paper, the u-invariant of the formallyreal field will be denoted by u(K). That also leads us to the following corollary.

Corollary 3.15. Let K be a formally real field and let u(K) <∞. Then u(K)is even.

Theorem 3.16. Let γ be a binary form and let ϕ be a form of dimension morethan or equal to 2. If the form γ · ϕ is isotropic, then the form ϕ contains abinary form β such that γ · β = 0 ∈W (K).

Proof. Let ϕ be a quadratic form such that dimϕ ≥ 2 and let β be a binarysubform of the form ϕ. Note that if the form ϕ is isotropic, as a subform β wecan put β = 〈1,−1〉. So, let ϕ be an anisotropic form and let γ = 〈s, t〉, wheres, t ∈ K, be a binary form. Suppose that the form γ · ϕ is isotropic. By theassumption, the form

γ · ϕ ∼= 〈s〉ϕ⊥〈t〉ϕis isotropic, so sx + ty = 0 for some x, y ∈ DK(ϕ). Let ϕ ∼= 〈x〉⊥ϕ1 for someform ϕ1. Then, we get the following equality y = xw2 + z, where w ∈ K and

16

z ∈ DK(ϕ1) ∪ {0}. Let us consider the following two cases.

(1) If z = 0, then y = xw2 for w ∈ K. Therefore, x and y represent thesame element of the group of square classes K/K2. Hence, and by the equalitysx = −ty, it follows that the form γ is hyperbolic. So, in this case, as β we canput any binary subform of the form ϕ.

(2) Suppose that z 6= 0. Then, the form ϕ1 can be represented as ϕ1∼=

〈z〉⊥ϕ2, for suitable form ϕ2. Therefore,

ϕ ∼= 〈x〉⊥ϕ1∼= 〈x, z〉⊥ϕ2.

So, the form ϕ consists the binary subform β ∼= 〈x, z〉. Note that, since y =xw2 + z, then y ∈ DK(x, z). By Theorem 2.10 we have 〈x, z〉 ∼= 〈y, xyz〉.Moreover, the following sequence of isometries takes place:

〈−t〉 · β ∼= 〈−ty,−ty · xz〉 ∼= 〈sx, sx · xz〉 ∼= 〈sx, sz〉 ∼= 〈s〉 · β.

So, 〈s〉 · β ∼= 〈−t〉 · β. And hence, 〈s, t〉 · β = γ · β = 0.

By the above theorem we get the following corollary.

Corollary 3.17. Let γ be a binary form such that γ � 〈1,−1〉 and let ϕ be anarbitrary quadratic form. If γ · ϕ = 0 ∈ W (K), then dimϕ = 2r for some r,and there exists an isometry

ϕ ∼= β1⊥ . . .⊥βr,

where β1, . . . , βr are binary forms such that γ · βi = 0 for i = 1, . . . , r.

Proof. Let γ be a binary form such that γ � 〈1,−1〉 and let ϕ be an arbitraryform. Suppose that γ · ϕ = 0. By induction on dimϕ, we will show the thesis.

Notice that the case when dimϕ = 1 is impossible, because γ is not ahyperbolic plane. So, γ〈a〉 6= 0 for every a ∈ K.

In the case when dimϕ = 2 the thesis is obvious. So, assume that dimϕ > 2.Since γ ·ϕ = 0 ∈W (K), so the form γ ·ϕ is isotropic. By the previous theorem,it follows that ϕ = β1⊥ϕ1, where dimβ1 = 2 and γ · β1 = 0. Therefore, wehave γ · ϕ = γ · ϕ1, where dimϕ1 = dimϕ − 2 < dimϕ. By Induction Thesis,it follows that ϕ1

∼= β2⊥ . . .⊥βr, where β2, . . . , βr are binary forms such thatγ · βi = 0 for i = 2, . . . , r. Hence, we have:

ϕ = β1⊥ . . .⊥βr.

So, by induction, we get the thesis.

By Theorem 3.16 and the previous corollary, it follows the case when γ ∼=〈1, 1〉, which will be useful in the later considerations.

Proposition 3.18. Let ϕ be a quadratic form over a field K. Then, there existr ≥ 0, binary forms β1, . . . , βr over K and a form ϕ0 over K such that

ϕ ∼= β1⊥ . . .⊥βr⊥ϕ0,

where 2βi = 0 for i = 1, . . . , r and either 2ϕ0 is anisotropic, or else dimϕ = 1.

17

Proof. We will induct on the dimension m of the form ϕ. If dimϕ = 1, existenceof the decomposition is obvious. Assume that dimϕ ≥ 2. If the form 2ϕ isanisotropic, then we put r = 0 and ϕ = ϕ0.

Now assume that 2ϕ = 〈1, 1〉ϕ is isotropic. Then, by Theorem 3.16, theform ϕ contains a binary subform β1 such that 2β1 = 0. So, ϕ = β1⊥ϕ1 forsome form ϕ1. Hence, 2ϕ = 2β1⊥2ϕ1, and in the Witt ring of K we have2ϕ = 2ϕ1. Naturally, dimϕ1 < dimϕ. So, by Induction Thesis, the form ϕ1 hasa decomposition ϕ1 = β2⊥ . . .⊥βr⊥ϕ0, where 2βi = 0 for i = 1, . . . , r and eitherthe form 2ϕ0 is anisotropic, or else dimϕ0 = 1. Then ϕ = β1⊥ . . .⊥βr⊥ϕ0, asdesired.

Now, let us recall that by a pythagorean field we mean a field K such thatevery element a ∈ K is a sum of two squares of elements from K. It is easy toprove that if K is pythagorean field, then u(K) ≤ 1 (see [2]).

Presented in the previous paragraph Kneser’s theorem concerns only thenonreal case. Therefore, we shall generalize Theorem 3.12 to the formally realcase. But first, let us prove the following theorem.

Theorem 3.19. Let K be an arbitrary field such that u = u(K) < ∞ and letϕ ∈Wt(K) be any u-dimensional anisotropic form. Then

V (ϕ) ≥ u.Proof. Suppose that u = u(K) < ∞. Let ϕ be a u-dimensional anisotropictorsion form. If u(K) = 1, then K is a quadratically closed field and thesisis obvious. So, we may assume that u = u(K) ≥ 2. Therefore, K is not apythagorean field. Let

ϕ ∼= β1⊥ . . .⊥βr⊥ϕ0

be a decomposition of the form ϕ such as in Proposition 3.18. We will showthat V (ϕ) ≥ u. The proof will be presented in the following four steps.

(1) As already noticed either 2ϕ0 is anisotropic, or else dimϕ0 = 1. In bothcases dim 2ϕ0 ≤ u. And hence,

4r = 2 · 2r = 2 · (dimϕ− dimϕ0) ≥ 2u− u = u.

(2) We shall show that the sets D(β1), . . . , D(βr), D(ϕ0) are mutually dis-joint subsets of D(ϕ). Indeed, reasoning by contradiction, suppose that thereexists an element x ∈ D(βi) ∩D(βj) for i 6= j. Let βi ∼= 〈x, y〉 for some y ∈ Kand let βj ∼= 〈x, z〉 for some z ∈ K. Moreover, we have the following isometry:

βi ∼= 〈−1〉βi ∼= 〈−x,−y〉.Considering the form βi⊥βj we get

βi⊥βj ∼= 〈−x,−y〉⊥〈x, z〉 ∼= 〈1,−1〉⊥〈−y, z〉.So, the form βi⊥βj is isotropic, a contradiction. Similarly, the sets D(βi) andD(ϕ0) are disjoint.

(3) Assume that s(K) ≤ 2. We will show that V (ϕ) ≥ u. Let ϕ0∼=

〈a2r+1, . . . , au〉. Notice that the elements a2r+1, . . . , au represent different squareclasses. Indeed, if aiK2 = ajK2 for i 6= j, then

βr+1 = 〈ai, aj〉 ∼= 2〈ai〉.

18

Therefore 2βr+1 = 0, which contradicts with the choice of r. And hence, itfollows the inequality:

V (ϕ0) ≥ dimϕ0 = u− 2r.

Now, we will show that V (βi) ≥ 2 for every 1 ≤ i ≤ r. Let βi ∼= 〈x, y〉. IfxK2 6= yK2, then of course V (βi) ≥ 2. So, suppose that xK2 = yK2. In thiscase βi ∼= 〈x, x〉. Since K is not a pythagorean field, so there exist elementsa, b ∈ K such that a2 + b2 is not a square. Hence, the set D(βi) contains atleast x and x(a2 + b2), where x and x(a2 + b2) represent distinct square classes.Therefore, also in this case V (βi) ≥ 2. Hence, and by the step (2), we have thefollowing sequence of inequalities:

V (ϕ) ≥r∑

i=1

V (βi) + V (ϕ0) ≥ 2r + (u− 2r) = u.

(4) Now assume that s(K) > 2. We shall show that V (βi) ≥ 4 for everyi = 1, . . . , r. Let βi ∼= 〈x, y〉. Since 4〈1〉 6= 0 and 2βi = 0, so xK2 6= yK2.Therefore, the set D(βi) contains elements x, y,−x,−y, which represent fourdistinct square classes. So, V (βi) ≥ 4.

Using the fact that V (βi) ≥ 4, we shall show that V (ϕ) ≥ u + 2. For thispurpose, let us consider the following two cases.

(i) 2r = uThen ϕ ∼= β1⊥ . . .⊥βr. Hence, and by the step (2), we get the following sequenceof inequalities:

V (ϕ) ≥r∑

i=1

V (βi) ≥ 4r = 2u ≥ u+ 2.

(ii) 2r < uAs already noticed, K is not a pythagorean field. So, if dimϕ0 ≥ 2, thenV (ϕ0) ≥ 2. By the steps (1) and (2) we get the inequality:

V (ϕ) ≥r∑

i=1

V (βi) + V (ϕ0) ≥ 4r + 2 ≥ u+ 2.

While, if dimϕ0 = 1, then u is an odd number. So, by the step (1) it followsthat 4r ≥ u+ 1. Finally,

V (ϕ) ≥ 4r + V (ϕ0) ≥ 4r + 1 ≥ u+ 2.

Now, by the above theorem, we may extend Kneser’s Theorem 3.12 to thecase of arbitrary fields.

Corollary 3.20. Let K be an arbitrary field. Then u(K) ≤ q(K).

Proof. We may assume that q(K) < ∞. Then, the Witt ring W (K) is finitelygenerated abelian group. The Structure Theorem for Abelian Groups statesthat every finitely generated abelian group G is isomorphic to a direct sum

Zn1 ⊕ . . .⊕ Znk⊕ Zn,

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where k ≥ 0, n ≥ 0 and n1, . . . , nk are powers of prime numbers (see [1]).Furthermore, the values of n, n1, . . . , nk are, up to rearranging the indices,uniquely determined by G. Therefore, we conclude that the torsion subgroupWt(K) of the Witt ring W (K) is isomorphic to a group Zn1 ⊕ . . . ⊕ Znk

, andhence also finite. So, the set of dimensions of all anisotropic torsion forms isfinite, which implies that u(K) <∞. The inequality u(K) ≤ q(K) follows fromTheorem 3.19.

Corollary 3.21. Let K be an arbitrary field and let s(K) > 2 and u(K) > 1.Then,

V (ϕ) ≥ u+ 2.

Proof. The proof follows from the step (4) of Theorem 3.19.

4 Conclusion

This paper is based on my Master Thesis, [3], written by supervision of dr hab.Alfred Czoga la. Apart from notions and theorems presented in the paper, wealso introduce a definition of system u-invariant as a u-invariant of the systemof n quadratic forms.

My Master Thesis concerns many examples of u-invariants of selected fields.Namely, we prove that u-invariant of a finite field is equal to 2. By Tsen–LangTheorem, we prove that u-invariant of a field of transcendence degree 1 overan algebraically closed field is also equal to 2. Moreover, we discuss completediscretely valuated field F with residue class field K of characteristic differentfrom 2, and prove that u(F ) = 2 ·u(K). As an example of that field we give thefield of formal Lauren series K((t)) and show that u(C((t1))((t2)) . . . ((tn))) =2n.

A significant part of my Master Thesis is devoted to u-invariant of finiteextensions. We prove that if L is a finite extension over a field K, then u(L) ≤12 (n+1) ·u(K). Furthermore, we show that if K is an arbitrary nonreal field andL = K(

√a), then u(L) ≤ 3

2 · u(K). First, we prove this theorem by propertiesof system u-invariant, and then, it is also proven by construction of Scharlau’sTransfer for finite extensions.

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References

[1] Browkin, J.: Theory of Fields (in Polish), Polish Scientific Publishers PWN,Warsaw 1977

[2] Elman, R.,Lam, T.Y.: Quadratic Forms and the u-Invariant. I, Mathema-tische Zeitschrift 131, 283–304 (1973), Springer-Verlag, 1973

[3] Kruszelnicka, M.: u-invariant (in Polish), Master Thesis, University ofSilesia, Katowice, 2009

[4] Lam, T.Y.: Introduction to Quadratic Forms over Fields, American Math-ematical Society, 2004

[5] Szymiczek, K.: Bilinear Algebra: An Introduction to the Algebraic Theoryof Quadratic Forms, Gordon and Breach Science Publishers, 1997

Institute of MathematicsUniversity of SilesiaBankowa 1440-007 Katowice, Poland

E-mail: gosia krusz@wp.pl

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