Pfister forms and K-theory of fields
Pfister forms and K-theory of fields
复制标题
普菲斯特形式和 K 场论
DOI：
10.1016/0021-8693(72)90054-3
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发表时间：
1972
影响因子：
0.9
通讯作者：
T. Lam
中科院分区：
文献类型：
--
作者：
R. Elman;T. Lam
本文的目的是研究CJI=（1，a，）@.@型二次型。（1，a，），我们称之为n重Pfister形式。这些形式首先由Pfister [7]系统地研究，他证明了Pfister形式的类本质上与所谓的强乘法形式的类一致（在特征不同于2的域F上）。对于n= 2，我们有CJI=（1，a，，aa，a&），它是四元数代数（-a，，-u，/F）的范数形式。一段时间以来，人们已经知道四元数代数（-al，-uJF）的同构类型和范数形式（1，a1，u2，urua）的等距类型是自然的一一对应关系，因此一个决定另一个，反之亦然。然而，对于n> 2的n重Pfister形式，迄今为止还没有分类理论。在文献[4]中，Milnor用Stiefel-Whitney类的方法证明了n重Pfister型y=（1，a，）@.（1，an）确定不变量I（-1）t-nZ（-u，）. 2（-Q）在代数K-群K $中，其中t= 2”-l.本文的主要定理是证明I（-ul）Z（-a，）Ek，F是上述n重Pfister型v的等距型的完全不变量（见定理3.2）。在证明这个定理中使用的技巧也有各种应用到Milnor [4]提出的一个问题，即k，F是否同构于ImF/IVF，其中I（F）表示Witt环W（F）中所有偶数维形式的理想。特别地，我们将能够证明，如果k，F最多有64个元素，情况确实如此。在第一节中，我们建立了本文的基本记号，回顾了二次型的一些熟悉的事实，然后建立了2-重Pfister型的一些基本性质。定理1.8关于
The object of this paper is to investigate quadratic forms of the type CJI=(1, a,)@...@(1, a,), which we shall call n-fold Pfister forms. These forms were first studied systematically by Pfister [7], who showed that the class of Pfister forms essentially coincides with the class of the so-called strongly multiplicative forms (over a field F of characteristic different from two). For n= 2, we have CJI=(1, a,, aa, a&, which is the norm form of the quaternion algebra (-a,,-u,/F). It has been known for some time that the isomorphism type of the quaternion algebra (-al,-uJF) and the isometry type of the norm form (1, a,, u2, urua) are in natural one-to-one correspondence, so that one determines the other, and vice versa. For n-fold Pfister forms with n> 2, however, no classification theory has been available so far. In his paper [4], by the method of Stiefel-Whitney classes, Milnor showed that the n-fold Pfister form y=(1, a,)@...@(1, an) determines an invariant I (-l) t-nZ (-u,)... 2 (-Q in the algebraic K-group K $, where t= 2”-l. Our main theorem in this paper is to establish that I (-ul) Z (-a,) E k, F is a complete invariant of the isometry type of the n-fold Pfister form v above (see Theorem 3.2). The techniques used in the proof of this theorem have also various applications to a question raised by Milnor [4], asking whether k, F is isomorphic to ImF/IVF, where I (F) denotes the ideal of all even-dimensional forms in the Witt ring W (F). In particular, we will be able to show that this is indeed the case, if k, F has at most 64 elements. In the first section, we set up the basic notations in this paper, review some familiar facts about quadratic forms, and then establish some elementary properties of 2-fold Pfister forms. Theorem 1.8 about the relationship between