Permuting the roots

Permuting the roots of univariate polynomials whose coefficients depend on parameters

A key characteristic of a family of polynomials is their Galois group, invented to determine whether these equations can be solved algebraically with an explicit, universal formula for their roots. We show that, remarkably, a Galois group can be computed for any family of polynomials with indeterminate coefficients. This allows us, for instance, to find all typical rational functions that are solvable by radicals.