We study the partial ordering on isomorphism classes of central simple algebras over a given field F, defined by setting \(A_1 \le A_2\) if \(\deg A_1 = \deg A_2\) and every étale subalgebra of \(A_1\) is isomorphic to a subalgebra of \(A_2\), and generalizations of this notion to algebras with involution. In particular, we show that this partial ordering is invariant under passing to the completion of the base field with respect to a discrete valuation, and we explore how this partial ordering relates to the exponents of algebras.

中文翻译：

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具有公共子域的除法代数

我们研究给定域F上中心简单代数的同构类的偏序，通过设置\(A_1 \le A_2\) if \(\deg A_1 = \deg A_2\)和\(A_1\ )与\(A_2\)的子代数同构，并且将此概念推广到具有对合的代数。特别是，我们证明了这种偏序在传递到关于离散估值的基域完成时是不变的，并且我们探索了这种偏序如何与代数的指数相关。