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首页> 外文期刊>Journal of algebra and its applications >On the classification of centrally finite alternative division rings satisfying algebraic closure conditions
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On the classification of centrally finite alternative division rings satisfying algebraic closure conditions

机译：关于满足代数封闭条件的中心有限交替除环的分类

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摘要

In this paper, we prove that the rings of quaternions and of octonions over an arbitrary real closed field are algebraically closed in the sense of Eilenberg and Niven. As a consequence, we infer that some reasonable algebraic closure conditions, including the one of Eilenberg and Niven, are equivalent on the class of centrally finite alternative division rings. Furthermore, we classify centrally finite alternative division rings satisfying such equivalent algebraic closure conditions: up to isomorphism, they are either the algebraically closed fields or the rings of quaternions over real closed fields or the rings of octonions over real closed fields.
机译：在本文中，我们证明了在任意实数封闭域上的四元数环和八元环在代数上是用Eilenberg和Niven表示的。结果，我们推断出一些合理的代数闭合条件，包括Eilenberg和Niven之一，在中心有限的另类划分环上是等效的。此外，我们对满足此类等效代数闭合条件的中心有限替代除环进行分类：直到同构，它们是代数闭合域或实闭合域上的四元数环或实闭合域上的八元环。

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