Symmetric bilinear forms and quadratic forms
Symmetric bilinear forms and quadratic forms
复制标题
对称双线性形式和二次形式
DOI：
10.1016/0021-8693(72)90094-4
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发表时间：
1972
影响因子：
0.9
通讯作者：
C. Sah
中科院分区：
文献类型：
--
作者：
C. Sah
在171，：\lilnor关联到每个域F一个分次和分次交换的I-ing K F。I 'he因子环k.：F~-1K，F？K，F则是域F上的分次交换代数a，有2个元素。当设F是：I gl：）haliicld或它的局部完备之一，但特征不是2，k时，F与F的二次倒数律有关。这就完成了各向异性二次型的L-LYitt环。在[8]中，iLiilnor证明了，不管E '的特征如何，P总是与各向异性对称半线性形式的环有关. Hn~ vwer，用于全局或局部字段1。特征2，k，F和二次倒数之间的关系，F的伊恩（在Hraucr群的意义上）并不明显。本文的主要目的是从对称双线性型中分离形式二次型。这在第I节中完成，我们检查在该设置中的多个ezan-l-les。特别地，从特征为2的有限向量空间到环Z; 4 Z的二次Inill>的概念如introdaccd 11）。我<.二. [3]第三个是肛门！xd在lkample I中有一些细节。4.\\：r tlic. 1)特征为2 in.%的场的二次倒数2002年。二次回归律与二次回归律相联系，其中：ns为周期。在第三节中，我们讨论了G_（？）应注意：1）在各方面，我们在scx~ ernl临界步骤上具有不可分离性。I：或特征不为2的fkld，我们的'I% cc~ rem 2的countcrpxt为0！3cn。
In 171,:\lilnor associated to each field F a graded and graded commutative I-ing K F.‘I’he factor ring k.: F~-1 K, F? K, F is then a graded commutative algclxa over the field F, with 2 elements.\Vhen one assumes that F is either: I gl:) hal iicld or one of Its local completions, but of characteristic not 2, k, F is related to the qlladratic reciprocit>-law of F. This can he accomplished througll tile LYitt ring of anisotropic quadratic forms. In [8], iLIilnor showed tha;/rf P is always related to the\\‘itt ring of anisotropic symmetric hilincal forms, regardless of the characteristic of E’. Hn~ vwer, for global or local fields 1.’of characteristic 2, the connection between k, F and the quadratic reciprocit!, Ian of F (in the sense of Hraucr group) is not in evidence. The main purpose of the present paper is to separate formally quadratic forms from symmetric bilinear forms. This is done in Section I\Ve examine a numlwr of ezan~ l~ les in this setting. In particular, the notion of a quadratic Inill> from a finite vector space of characteristic 2 into the ring Z; 4Z as introdaccd 11). I<. II. Erown [3] is anal! xd in some detail in lkample I. 4.\\: r tlic. 1) rvstorc quadratic reciprocitv f0r fields of characteristic 2 in.% ction 2. Iic~ ugliI~-, quadratic reciprucit\ laws arc connected with quadratic for: ns as cycted. In Section 3 we eraminc some of the relations with G&is cohomc~ lo, g;. It should 1)~ ohservcd that we hare cuploitcd inseparability in scx~ ernl cr-itical steps among the pronfs. I: or fklds of characteristic not 2, the countcrpxt of our ‘I% cc~ rem 2 is 0! 3cn.