Abstract. We show that if a system of differential equations has an elementary first integral (i.e. a first integral expressible in terms of exponentials, logarithms and algebraic functions) then it must have a first integral of a very simple form. This unifies and extends results of Mordukhai-Boltovski, Ritt and others and leads to a partial algorithm for finding such integrals.

This paper gives a corollary to Schanuel's conjecture that indicates when an exponential or logarithmic constant is transcendental over a given field of constants. The given field is presumed to have been built up by starting with the rationals Q with π adjoined and taking algebraic closure, adjoining values of the exponential function or of some fixed branch of the logarithmic function, and then repeating these two operations a finite number of times.

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Numerous investigations on integration in finite form of Abelian integrals are based on theorems of Liouville [6,7,8] and Abel [1].