Moscow Mathematical Journal
Volume 19, Issue 2, April–June 2019 pp. 329–341.
Integrability in Finite Terms and Actions of Lie Groups
According to Liouville's Theorem, an idefinite integral of an
elementary function is usually not an elementary function. In
these 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
these notes is dedicated to the second step dealing 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.F. Ritt. 2010 Math. Subj. Class. 12H05
Authors:
Askold Khovanskii (1)
Author institution:(1) University of Toronto, Department of Mathematics, Toronto, ON M5S 2E4, Canada
Summary:
Keywords: Liouville's theorem on integrability in finite terms, differential Galois group, elementary function
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