Prime rational functions
Let $f(x)$ be a complex rational function. We study conditions under which $f(x)$ cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that $f(x)$ is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we also derive some conditions for the case of complex polynomials.