素数次非结合循环代数的参数化
A parametrization of nonassociative cyclic algebras of prime degree
AI总结：
本文参数化了特征非2域上的非结合四元数代数及含本原单位根时奇素数次非结合循环代数的同构类，并给出非阿基米德局部域上的显式结果。
AI中文摘要：
我们确定并显式参数化了特征不为2的域上非结合四元数代数的同构类，以及当基域包含一个本原 $m$ 次单位根时奇素数次非结合循环代数的同构类。在此过程中，我们证明了任意两个这样的代数只有在循环域扩张和所选 Galois 群生成元相同时才可能同构。作为应用，我们给出了局部非阿基米德域 $F$ 上素数次非结合循环代数的参数化，在对剩余特征的温和假设下，该参数化是完全显式的。特别地，这使我们对非阿基米德局部域上这一重要的非结合四元数代数类在同构意义下有了丰富的理解。
英文摘要：
We determine and explicitly parametrize the isomorphism classes of nonassociative quaternion algebras over a field of characteristic different from two, as well as the isomorphism classes of nonassociative cyclic algebras of odd prime degree when the base field contains a primitive $m$th root of unity. In the course of doing so, we prove that any two such algebras can be isomorphic only if the cyclic field extension and the chosen generator of the Galois group are the same. As an application, we give a parametrization of nonassociative cyclic algebras of prime degree over a local nonarchimedean field $F$, which is entirely explicit under mild hypotheses on the residual characteristic. In particular, this gives a rich understanding of the important class of nonassociative quaternion algebras up to isomorphism over nonarchimedean local fields.
