Variations on the theme of solvability by radicals

We discuss the problem of representability and nonrepresentability of algebraic functions by radicals. We show that the Riemann surfaces of functions that are the inverses of Chebyshev polynomials are determined by their local behavior near branch points. We find lower bounds on the degrees of equations to which sufficiently general algebraic functions can be reduced by radicals. We also begin to classify rational functions of prime degree whose inverses are representable by radicals.