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Daniel Krashen

机译:丹尼尔·克拉申

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摘要

A conjecture of Amitsur states that two Severi-Brauer varieties are birationally isomorphic if and only if the underlying algebras are the same degree and generate the same cyclic subgroup of the Brauer group. It is known that generating the same cyclic subgroup is a necessary condition, however it has not yet been shown to be sufficient. In this paper we examine the case where the algebras have a maximal subfield $K/F$ of degree $n$ with Galois closure $E/F$ whose Galois group is of the form $C_n times H$, where $E^H = K$ and $|H|$ is prime to $n$. For such algebras we show that the conjecture is true for certain cases of $n$ and $H$. In particular we prove the conjecture in the case that $G$ is a dihedral group of order $2p$, where $p$ is prime.
机译:Amitsur的一个猜想指出,当且仅当下面的代数具有相同的度数并产生相同的Brauer群的循环子群时,两个Severi-Brauer变体才是同构的。已知产生相同的循环亚组是必要条件,但是尚未证明是足够的。在本文中,我们研究了代数具有最大子域$ K / F $且度数为$ n $且Galois闭包$ E / F $的情况,其Galois组的形式为$ C_n rtimes H $,其中$ E ^ H = K $,$ | H | $是$ n $的质数。对于这样的代数,我们证明在$ n $和$ H $的某些情况下该猜想是正确的。特别是在$ G $是阶数为$ 2p $的二面体组(其中$ p $是质数)的情况下,我们证明了这种猜想。

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