域论与某些伽罗瓦群的上同调
Field theory and the Cohomology of Some Galois Groups
摘要：
证明了域F的两个算术有效扩张重合当且仅当Witt环WF是群环Z/n[G].进一步，利用2-群的伽罗瓦群的模平方，建立了一个类似于Hilbert定理90的定理，并证明了F的二次闭包的伽罗瓦群的上同调维数，某个模在伽罗瓦群上的滤子的长度，域的平方类群的Z/2上的维数对许多有趣的域族成立。最后，我们讨论了一个特殊的伽罗瓦群在拓扑背景下的上同调。
英文摘要：
We prove that two arithmetically significant extensions of a field F coincide if and only if the Witt ring WF is a group ring Z/n[G]. Furthermore, working modulo squares with Galois groups which are 2-groups, we establish a theorem analogous to Hilbert's Theorem 90 and show that an identity linking the cohomological dimension of the Galois group of the quadratic closure of F, the length of a filtration on a certain module over a Galois group, and the dimension over Z/2 of the square class group of the field holds for a number of interesting families of fields. Finally we discuss the cohomology of a particular Galois group in a topological context.
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