Rational constants of monomial derivations

We present some general properties of the field of constants of monomial derivations of k(x1,…,xn), where k is a field of characteristic zero. The main result of this paper is a description of all monomial derivations of k(x,y,z) with trivial field of constants. In this description a crucial role plays the classification result of Moulin Ollagnier for Lotka–Volterra derivations with strict Darboux polynomials. Several applications of our description are also given in this paper.