Relative Brauer groups II.
Relative Brauer groups II.
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DOI：
10.1515/crll.1981.328.39
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发表时间：
1981
期刊：
影响因子：
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通讯作者：
M. Schacher;B. Fein;W. Kantor
中科院分区：
文献类型：
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作者：
M. Schacher;B. Fein;W. Kantor
设L^K是域，L/K的相对Brauer群B（L/K）表示K的Brauer群B（K）的子群，该群由被L分裂的有限维中心单AT-代数的Brauer类组成。本文继续了文献[10]中对B（L/K）结构的研究。设L=>K是整体域，p是素数整除[L '. K]. (By整体域，我们指的是有限域上的一个单变量代数数域或代数函数域。）如[10]命题4的证明所示，如果L是K上的伽罗瓦，则B（L/K）的准分量B（L /K）p是无限的.然而，[10]的例子1表明，如果L不是K上的伽罗瓦，则这不一定成立。这就引出了下面的自然问题：是否存在全局域L => K，L Φ K，其中B（L/K）是有限的？我们在§ 3中证明了这样的域的存在性等价于作用在集合Ω上的有限传递置换群G的存在性，|Ω|>1，且G的所有素数幂阶非平凡元在Ω上有不动点.在有限单群的分类是完备的假设下，我们在§ 2中给出了一个证明，证明不存在这样的对（G，Ω）。
Let L^K be fields and let B(L/K), the relative Brauer group of L/K, denote the subgroup of the Brauer group B (K) of K consisting of those Brauer classes of fmite dimensional central simple AT-algebras which are split by L. In this paper we continue the investigation of the structure of B(L/K) begun in [10]. Let L=>K be global fields and let p be a prime dividing [L'.K]. (By a global field we mean either an algebraic number field or an algebraic function field in one variable over a finite field.) As shown in the proof of [10], Proposition 4, the /?-primary component B (L/K) p of B (L / K) is infinite if L is Galois over K. Example l of [10] shows, however, that this need not hold if L is not Galois over K. This raises the following natural question: do there exist global fields L => K, L Φ K, with B(L/K) finite? We show in § 3 that the existence of such fields is equivalent to the existence of a finite transitive permutation group G acting on a set Ω, |Ω|>1, with the property that all nontrivial elements of G of prime power order have fixed points on Ω. Under the assumption that the classification of the finite simple groups is complete, we sketch a proof in § 2 that no such pair (G, Ω) exists.