Abstract

For each field $K$ of characteristic $0$, we determine all degree-$n$ rational functions $f(X) \in K(X)$ for which the Galois group of the Galois closure of $K(x)/K(f(x))$ is not $A_n$ or $S_n$.  For many applications, this means determining all rational functions over $K$ which behave differently from a typical rational function of the same degree.  We give applications to value distribution of meromorphic functions, near-injectivity of rational functions over number fields, and bijectivity of rational functions on subgroups of the multiplicative group of a finite field.  The proofs rely on various results classifying primitive groups with additional properties, and the topics lead to new types of questions about primitive groups. Abstract (in regular text): For each field K of characteristic 0, we determine all degree-n rational functions f(X) in K(X) for which the Galois group of the Galois closure of K(x)/K(f(x)) is not A_n or S_n.  For many applications, this means determining all rational functions over K which behave differently from a typical rational function of the same degree.  We give applications to value distribution of meromorphic functions, near-injectivity of rational functions over number fields, and bijectivity of rational functions on subgroups of the multiplicative group of a finite field.  The proofs rely on various results classifying primitive groups with additional properties, and the topics lead to new types of questions about primitive groups.