$p$-Central Subspaces of Central Simple Algebras

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.