Defining a function of one
variable to be elementary if it has an explicit representation in terms of a finite
number of algebraic operations, logarithms, and exponentials, Liouville’s theorem in
its simplest case says that if an algebraic function has an elementary integral then
the latter is itself an algebraic function plus a sum of constant multiples of
logarithms of algebraic functions. Ostrowski has generalized Liouville’s results to
wider classes of meromorphic functions on regions of the complex plane
and J. F. Ritt has given the classical account of the entire subject in his
Integration in Finite Terms, Columbia University Press, 1948. In spite of
the essentially algebraic nature of the problem, all proofs so far have been
analytic. This paper gives a self contained purely algebraic exposition of the
problem, making a few new points in addition to the resulting simplicity and
generalization.
