The Witt ring kernel for a fourth degree field extension and related problems
The Witt ring kernel for a fourth degree field extension and related problems
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四度场扩展的维特环核及相关问题
DOI：
10.1016/j.jpaa.2009.04.015
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发表时间：
2010
影响因子：
0.8
通讯作者：
A. S. Sivatski
中科院分区：
文献类型：
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作者：
A. S. Sivatski
本文的第一部分是对任意一个特征不等于2的4次域扩张，用一个决定扩张的多项式的系数计算Witt环核。在下域不是形式上真实的的情况下，证明了其基本理想的任意幂n与Witt环核的交由n重Pfister形式生成.第二部分作为主要结果的应用，给出了四元数代数和双四元数代数张量积有零因子的一个判别准则。并对三个四元数代数解决了类似的问题。最后一部分我们得到了关于二面体Galois域扩张的某些精确Witt群序列。这些结果在很大程度上依赖于Positselski的一些类似的上同调结果，以及Voevodsky证明的Milnor猜想和指数为2的Bloch-Kato猜想。
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.