非交叉积定理部分内容的证明

arXiv - MATH - Rings and Algebras Pub Date : 2024-08-22 DOI:arxiv-2408.12711

Mehran Motiee

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引用次数: 0

摘要

1972 年，阿米瑟给出了非交叉积分代数的第一个例子。他的方法基于两个基本步骤：(1) 如果普分代数 $U(k,n)$ 是 $G$ 交叉积，那么 $k$ 上的每个度数为 $n$ 的分代数都应该是 $G$ 交叉积；(2) $k$ 上有两个分代数，它们的最大子域没有共同的伽罗瓦群。在本注中，我们给出了在 $\chr k\nmid n$ 和 $p^3|n$ 的情况下第二步的简短证明。

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A proof for a part of noncrossed product theorem

The first examples of noncrossed product division algebras were given by Amitsur in 1972. His method is based on two basic steps: (1) If the universal division algebra $U(k,n)$ is a $G$-crossed product then every division algebra of degree $n$ over $k$ should be a $G$-crossed product; (2) There are two division algebras over $k$ whose maximal subfields do not have a common Galois group. In this note, we give a short proof for the second step in the case where $\chr k\nmid n$ and $p^3|n$.

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arXiv - MATH - Rings and Algebras

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