Going up of the <Emphasis Type="Italic">u</Emphasis>-invariant over formally real fields

Let F be a field of characteristic not 2, and assume that F has finite reduced stability. Let K/F be any finite extension. We prove that if the general u-invariant u(F) is finite, then u(K) is finite. This article is based on part of the author’s Ph.D. thesis, written under the supervision of Richard Elman.     

On fields of u-invariant 4

This note is motivated by the problem of determining the u-invariant of a field F of characteristic different from two when it is known that A criterion is given to decide whether u(F) ≤ 4 in this situation. Received: 25 February 2005     

On the norm theorem for semisingular quadratic forms

The aim of this paper is to prove some results concerning the norm theorem for semisingular quadratic forms, i.e., those which are neither nonsingular nor totally singular. More precisely, we will give necessary conditions in order that an irreducible polynomial, possibly in more than one variable, is a norm ofa semisingular quadratic form, and we prove that our conditions are sufficient if the polynomial is given by a quadratic form which represents 1. As a consequence, we extend the Cassels-Pflster subform theorem to the case of semisingular quadratic forms.     

The Witt ring kernel for a fourth degree field extension and related problems

In the first part of this paper 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.In the second part as an application of the main result we give a criterion for the tensor product of quaternion and biquaternion algebras to have zero divisors. Also we solve the similar problem for three quaternion algebras.In the last part we obtain certain exact Witt group sequences concerning dihedral Galois field extensions. These results heavily depend on some similar cohomological results of Positselski, as well as on the Milnor conjecture, and the Bloch-Kato conjecture for exponent 2, which was proven by Voevodsky.     

The behavior of differential forms under purely inseparable extensions

Let F be a field of characteristic 2. In this paper we give a complete computation of the kernel of the homomorphism $H_2^{m+1}(F)?H_2^{m+1}(L)$ induced by scalar extension, where $L/F$ is a purely inseparable extension (of any degree), $H_2^{m+1}(F)$ is the cokernel of the Artin–Schreier operator $?:{\mathrm{\Omega }}_F^m?{\mathrm{\Omega }}_F^m/\mathrm{d}{\mathrm{\Omega }}_F^{m?1}$ given by: $x\frac{\mathrm{d}{x}_1}{x_1}\wedge ?\wedge \frac{\mathrm{d}{x}_m}{x_m}?(x^2?x)\frac{\mathrm{d}{x}_1}{x_1}\wedge ?\wedge \frac{\mathrm{d}{x}_m}{x_m}+\mathrm{d}{\mathrm{\Omega }}_F^{m?1}$, where ${\mathrm{\Omega }}_F^m$ is the space of absolute m-differential forms over F and d is the differential operator. Other related results are included.     

Pairs of quadratic forms over a quadratic field extension

Let F be a field of characteristic distinct from 2, $L=F(\sqrt{d})$ a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, $M_f$, $M_g$ their matrices. We say that the pair $(f,g)$ is a k-pair if there exist $S\in G{L}_n(L)$ such that all the entries of the $k\times k$ upper-left corner of the matrices $S{M}_f{S}^t$ and $S{M}_g{S}^t$ are in F. We give certain criteria to determine whether a given pair $(f,g)$ is a k-pair. We consider the transfer ${\mathrm{cor}}_{L(t)/F(t)}$ determined by the $F(t)$-linear map $s:L(t)\to F(t)$ with $s(1)=0$, $s(\sqrt{d})=1$, and prove that if $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a $[\frac{k+1}{2}]$-pair. If, additionally, the form $f+tg$ does not have a totally isotropic subspace of dimension $p+1$ over $L(t)$, we show that $(f,g)$ is a $(k?2p)$-pair. In particular, if the form $f+tg$ is anisotropic, and $\dim {\mathrm{cor}}_{L(t)/F(t)}{(f+tg)}_{an}\le 2(n?k)$, then $(f,g)$ is a k-pair.     

浅谈正定二次型的习题课设计

在高等代数的实二次型内容中,正定二次型占有特殊的地位.本文从概念的回顾、正定二次型与正定矩阵的判断、二次型正定及矩阵正定的性质、其它类型二次型四个方面来设计正定二次型的习题课,并通过具体例子说明例题、习题精选的原则.     

On split products of quaternion algebras with involution in characteristic two

The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution is shown to have an affirmative answer.     

Minimal weakly isotropic forms

In this article weakly isotropic quadratic forms over a (formally) real field are studied. Conditions on the field are given which imply that every weakly isotropic form over that field has a weakly isotropic subform of small dimension. Fields over which every quadratic form can be decomposed into an orthogonal sum of a strongly anisotropic form and a torsion form are characterized in different ways.     

Totally positive extensions and weakly isotropic forms

The aim of this article is to analyse a new field invariant, relevant to (formally) real fields, defined as the supremum of the dimensions of all anisotropic, weakly isotropic quadratic forms over the field. This invariant is compared with the classical u-invariant and with the Hasse number. Furthermore, in order to be able to obtain examples of fields where these invariants take certain prescribed values, totally positive field extensions are studied.     

u-Invariants for forms of higher degree

Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. We give a survey of both old and new results on these u-invariants.     

Sesquilinear forms over rings with involution

Many classical results concerning quadratic forms have been extended to Hermitian forms over algebras with involution. However, not much is known in the case of sesquilinear forms without any symmetry property. The present paper will establish a Witt cancellation result, an analogue of Springer’s theorem, as well as some local–global and finiteness results in this context.     

Generic points of quadrics and Chow groups

In this article we study the sufficient conditions for the k̅-defined element of the Chow group of a smooth variety to be k-rational (defined over k). For 0-cycles this question was addressed earlier. Our methods work for cycles of arbitrary dimension. We show that it is sufficient to check this property over the generic point of a quadric of sufficiently large dimension. Among the applications one should mention the uniform construction of fields with all known u-invariants.     

Hankel matrices and quadratic forms

A characterization of finite Hankei matrices is given and it is shown that such matrices arise naturally as matrix representations of scaled trace forms of field extensions and etale algebras. An algorithm is given for calculating the signature and the Hasse invariant of these scaled trace forms.