微分符号代数的分裂域

Common Splitting Fields of Symbol Algebras

We study the common splitting fields of symbol algebras of degree $p^m$ over fields $F$ of $\operatorname{char}(F)=p$. We first show that if any finite number of such algebras share a degree $p^m$ simple purely inseparable splitting field, then they share a cyclic splitting field of the same degree. As a consequence, we conclude that every finite number of symbol algebras of degrees $p^{m_0},\dots,p^{m_t}$ share a cyclic splitting field of degree $p^{m_0+\dots+m_t}$. This generalization recovers the known fact that every tensor product of symbol algebras is a symbol algebra. We apply a result …

Period and index, symbol lengths, and generic splittings in Galois cohomology

We use constructions of versal cohomology classes based on a new notion of "presentable functors," to describe a relationship between the problems of bounding symbol length in cohomology and of finding the minimal degree of a splitting field. The constructions involved are then also used to describe generic splitting varieties for degree 2 cohomology with coefficients in a commutative algebraic group of multiplicative type.

微分四元数代数的分裂

Splitting of differential quaternion algebras

本文研究带导子的四元数代数的微分分裂域问题，指出其未必在代数扩张上分裂，并利用 Riccati 方程的解给出分裂域超越次数的界。

Finite Splittings of Differential Matrix Algebras

It is well known that central simple algebras are split by suitable finite Galois extensions of their centers. A counterpart of this result was studied by Juan and Magid in the set up of differential matrix algebras, wherein Picard-Vessiot extensions that split matrix differential algebras were constructed. In this article, we exhibit instances of differential matrix algebras which are split by finite extensions. In some cases, we relate the existence of finite splitting extensions of a differential matrix algebra to the triviality of its tensor powers, and show in these cases, that orders of …

Linkage of Symbol $p$-Algebras of Degree 3

Given a field $F$ of characteristic $3$ and division symbol $p$-algebras $[α,β)_{3,F}$ and $[α,γ)_{3,F}$ of degree $3$ over $F$, we prove that if $α\text{dlog}(β)\wedge \text{dlog}(γ)$ is trivial in the Kato-Milne cohomology group $H_3^3(F)$ then the algebras share a common splitting field which is an inseparable degree 3 extension of either $F$ or a quadratic extension of $F$. In the special case of quadratically closed fields, if $α\text{dlog}(β)\wedge \text{dlog}(γ)=0$, then they share an inseparable degree 3 extension of $F$.

微分中心单代数

Differential Central Simple Algebras

本文综述了微分中心单代数的研究，回顾了微分子域和微分分裂域等关键概念，重点探讨了微分分裂域与经典微分伽罗瓦理论的联系，并提出了若干待解问题。