On the Inverse Problem in Differential Galois Theory
On the Inverse Problem in Differential Galois Theory
复制标题
论微分伽罗瓦理论中的反问题
DOI：
10.11588/heidok.00003085
复制
发表时间：
2002
影响因子：
3.1
通讯作者：
Julia Hartmann
中科院分区：
文献类型：
--
作者：
Julia Hartmann
微分伽罗瓦理论将多项式的伽罗瓦理论推广到微分方程。微分方程有分裂域（Picard-Vessiot展开式）的概念，微分伽罗瓦群是这个展开式的自同构群，这些自同构固定了基域并与导数交换。微分伽罗瓦群是基域常数域上的线性代数群。与经典情况类似，我们考虑一个反问题：在给定的微分域上，哪些线性代数群作为微分伽罗瓦群出现？本文的主要结果是：定义在代数闭域K上的每一个线性代数群，都以导数为d/dt的某Picard-Vessiot扩展的微分伽罗瓦群出现。
Differential Galois theory generalizes the usual Galois theory for polynomials to differential equations. There is the notion of a splitting field (Picard-Vessiot extension) of a differential equation, and the differential Galois group is the group of automorphisms of this extension which fix the base field and commute with the derivation. Differential Galois groups are linear algebraic groups over the field of constants of the base field. In analogy to the classical situation, one considers the inverse problem: Which linear algebraic groups occur as differential Galois groups over a given differential field? The main result of this thesis is the following theorem: Every linear algebraic group defined over the algebraically closed field K occurs as the differential Galois group of some Picard-Vessiot extension of K(t) with derivation d/dt.