A Brioschi quintic is a form of the general quintic whose coefficients are integer polynomial functions in a single variable. Viewed as an algebraic curve, it is a smooth, irreducible curve admitting of a rational parametric representation. Thus Brioschi quintics solvable by radicals with one rational root are numerous, however, irreducible quintics may also be solvable by radicals. In this paper, solvable Brioschi quintics with rational coefficients are described in full. Remarkably, any Brioschi quintic with integer coefficients is unsolvable by radicals.

quintic, Galois group, Diophantine equations.