Fast decomposition of polynomials with known Galois group

Let f(X) be a separable polynomial with coefficients in a field K, generating a field extension M/K. If this extension is Galois with a solvable automorphism group, then the equation f(X) = 0 can be solved by radicals. The first step of the solution consists of splitting the extension M/K into intermediate fields. Such computations are classical, and we explain how fast polynomial arithmetic can be used to speed up the process. Moreover, we extend the algorithms to a more general case of extensions that are no longer Galois. Numerical examples are provided, including results obtained with our implementation for Hilbert class fields of imaginary quadratic fields.