For a polynomial f in C[X], let G be the Galois group of the Galois closure of the field extension C(X)|C(f(X)). We classify the groups G in the indecomposable case. For polynomials with rational coefficients there are, besides four infinite series, only three more ``sporadic" examples. In the Appendix we reprove the classical Theorems of Ritt about decompositions of polynomials using the group-theoretic setup.
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