(Prior to publication, this paper should be cited as arXiv:1405.4753.)

We examine the different ways of writing a cover of curves h : C→D over a field K as a composition of covers h = h₁ o h₂ o ... o h_n, where each h_i has degree at least 2 and cannot be written as the composition of lower-degree covers. Write Mon(h) for the monodromy group of h, namely the Galois group of the Galois closure of the function field extension K(C)/K(D). We show that if Mon(h) has a transitive abelian subgroup then the sequence (deg(h_i)_1≤i≤n is uniquely determined (up to permutation) by h, so in particular the length n of the decomposition is uniquely determined. We prove analogous conclusions for the sequences (Mon(h_i))_1≤i≤n and (Aut(h_i))_1≤i≤n. Such a transitive abelian subgroup exists in particular when h is tamely and totally ramified over some point in D(K), and also when h is a morphism of one-dimensional algebraic groups (or a coordinate projection of such a morphism). Thus, for example, our results apply to decompositions of polynomials of degree not divisible by char(K), additive polynomials, elliptic curve isogenies, and Lattès maps. In this way we obtain new results in each of these situations, while also obtaining a common generalization of results of Ritt, Ore, Beardon–Ng, and Müller–Zieve.

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