Let K be a field. The monodromy group of a rational function $r(X) = f(X)/g(X) \in K(X)$, i.e., the Galois group of $f(X) − tg(X)$ over $K(t)$, is an important object of study in problems from number theory, geometry, arithmetic dynamics, etc.
Classifying which finite groups occur as monodromy groups has been of great interest, since this knowledge helps reducing many arithmetic problems to pure group theory. The celebrated Guralnick-Thompson conjecture (1990; eventually proved by Frohardt and Magaard) asserts that apart from alternating and cyclic groups, only finitely many simple groups occur as composition factors of monodromy groups of rational functions over C (so-called "geometric" monodromy groups). In the case of functionally indecomposable $r(X)$, later work by Neftin, Zieve and others classified not only the "exceptional" groups, but actually the rational functions with exceptional monodromy group, assuming sufficiently large degree. In joint work in progress with Mueller, Neftin and Zieve, we reach a similar result for "arithmetic" monodromy groups. That is, we extend the above classification to arbitrary fields of characteristic zero. As a consequence, we also prove a generalization of the Guralnick-Thompson conjecture for arbitrary fields.