Includes the first serious use of R(iemann)'s E(xistence) T(heorem) on a problem of this type, a start of the monodromy method. Chebychev covering groups are dihedral and easy to characterize. So, RET was quick, but not essential here. Yet, Schur's Conjecture was special, and much easier, within Davenport's problem, and RET has proved essential for that. Still, by considering its analog for rational functions, the monodromy method connected to Serre's O(pen)I(mage)T(heorem) (UMStoryExc-OIT.html and GCMTAMS78.pdf) and, so, to modular curves. Further, using RET opened the territory to many other problems (especially see the complete story of Davenport's problem in UMStory.pdf).
Davenport's problem was essentially to classify polynomials over Q by their ranges on almost all residue class fields. The most general results, restricted to polynomials not composable (indecomposable) from lower degree polynomials, gave two very different conclusions: