有限项可积性与李群的作用
Integrability in Finite Terms And Actions of Lie Groups
摘要：
根据刘维尔定理，初等函数的不定积分通常不是初等函数。在本文中，我们讨论了这一说法和证明这一结果。通过邻接积分得到的扩张的微分伽罗瓦群并不决定该积分是否是初等函数。然而，刘维尔定理可以用微分伽罗瓦群来证明。第一步这样的证明是由阿贝尔建议。这一步与代数扩张及其有限伽罗瓦群有关。本笔记的重要部分是致力于第二步，其中涉及纯超越扩展和它们的伽罗瓦群是连接的李群。证明的思想可以追溯到J.Liouville和J.Ritt。
英文摘要：
According to Liouville's Theorem, an indefinite integral of an elementary function is usually not an elementary function. In this notes, we discuss that statement and a proof of this result. The differential Galois group of the extension obtained by adjoining an integral does not determine whether the integral is an elementary function or not. Nevertheless, Liouville's Theorem can be proved using differential Galois groups. The first step towards such a proof was suggested by Abel. This step is related to algebraic extensions and their finite Galois groups. A significant part of this notes is dedicated to a second step, which deals with pure transcendent extensions and their Galois groups which are connected Lie groups. The idea of the proof goes back to J.Liouville and J.Ritt.
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