标题： $p$-中心单代数的中心子空间 显示英文标题

标题： $p$-Central Subspaces of Central Simple Algebras

摘要： 我们以多种方式研究中心单代数，重点在于$p$-中心子空间的作用。 我的论文的第一部分致力于研究Clifford代数。 给定形式的标准Clifford代数是包含一个$p$-中心子空间的通用结合代数，其指数形式等于给定形式。 有一个古老的问题是这些代数在中心上是否具有有限秩表示，我和 Daniel Krashen 以及 Max Lieblich 一起成功地给出了肯定的答案。 该部分还介绍了并研究了Clifford代数结构的不同推广形式。 第二部分致力于研究给定中心单代数中的$p$-中心子空间，主要是次数为$p$的循环代数的张量积。 在结果中，我们证明$5$是包含标准生成元对的次数为4的循环代数的4-中心子空间维度的上限。 第三部分致力于链引理的研究。 链引理在中心单代数理论中很重要，因为它们是解决Brauer群字问题的一种方法。 我们证明了双四元数代数的链引理，包括特征2和特征不为2的情况，并证明了关于次数为$p$的循环代数的链引理的一些部分结果。 第四部分致力于理论的更计算方面的内容。 它包含关于四元数多项式方程和四元数矩阵的左特征值的结果。 显示英文摘要

摘要： We study central simple algebras in various ways, focusing on the role of $p$-central subspaces. The first part of my thesis is dedicated to the study of Clifford algebras. The standard Clifford algebra of a given form is the generic associative algebra containing a $p$-central subspace whose exponentiation form is equal to the given form. There is an old question as for whether these algebras have representations of finite rank over the center, and jointly with Daniel Krashen and Max Lieblich we managed to provide a positive answer. Different generalizations of the structure of the Clifford algebra are presented and studied in that part too. The second part is dedicated to the study of $p$-central subspaces of given central simple algebras, mainly tensor products of cyclic algebras of degree $p$. Among the results, we prove that $5$ is the upper bound for the dimension of 4-central subspaces of cyclic algebras of degree 4 containing pairs of standard generators. The third part is dedicated to chain lemmas. Chain lemmas are of importance in the theory of central simple algebras, because they form one approach to solving the word problem for the Brauer group. We prove the chain lemma for biquaternion algebras, both in characteristic 2 and characteristic not 2, and prove some partial results on the chain lemmas for cyclic algebras of degree $p$. The fourth part is dedicated to the more computational aspects of the theory. It contains results on quaternion polynomial equations and on left eigenvalues of quaternion matrices.