Quadratic forms and Pfister neighbors in characteristic 2
Quadratic forms and Pfister neighbors in characteristic 2
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特征 2 中的二次形式和 Pfister 邻域
DOI：
10.1090/s0002-9947-04-03461-0
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发表时间：
2004
影响因子：
1.3
通讯作者：
Ahmed Laghribi
中科院分区：
文献类型：
--
作者：
D. Hoffmann;Ahmed Laghribi
我们研究特征2的域上的菲斯特邻居及其特征，其中我们包括奇异形式的情况。我们给出了Fitzgerald定理的一个简化证明，该定理提供了一个判别准则，当非奇异二次型q在由q支配的形式φ的函数域上的双曲性方面与Pfister型相似时，该形式是双曲的。从这一点，我们得出了一个类似的结果的特点2由Knebusch说，在特点<$= 2，一个形式是一个Pfister邻居，如果它的各向异性部分在其自己的功能领域被定义在基础领域。我们的结果包括某些情况下的奇异形式，但我们也给出的例子表明，Knebusch的结果一般失败的奇异形式的特征2。作为应用，我们刻画了Knebusch意义下高度1的某些形式，其拟线性部分是小维数的.我们还发展了一些基本的理论完全奇异二次型。这是用来给一个新的解释的概念，一个标准的分裂塔的高度介绍了第二作者在较早的文件。
We study Pfister neighbors and their characterization over fields of characteristic 2, where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form q is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form φ which is dominated by q. From this, we derive an analogue in characteristic 2 of a result by Knebusch saying that, in characteristic ¬= 2, a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch's result generally fails in characteristic 2 for singular forms. As an application, we characterize certain forms of height 1 in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.