Solvability by radicals is in polynomial time

• Authors: Susan Landau;Gary Lee Miller
• Venue: STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
• Year: 1983

Every high school student knows how to express the roots of a quadratic equation in terms of radicals; what is less well-known is that this solution was found by the Babylonians a millenia and a half before Christ [Ne]. Three thousand years elapsed before European mathematicians determined how to express the roots of cubic and quartic equations in terms of radicals, and there they stopped, for their techniques did not extend. Lagrange published a treatise which discussed why the methods that worked for polynomials of degree less than five did not work for quintic polynomials [Lag], They require double exponential time. Through the years other mathematicians developed alternate algorithms all of which, however, remained exponential. A major impasse was the problem of factoring polynomials, for until the recent breakthrough of Lenstra, Lenstra, and Lovász [L3], all earlier algorithms had exponential running time. Their algorithm, which factors polynomials over the rationals in polynomial time, gave rise to a hope that some of the classical questions of Galois theory might have polynomial time solutions. Galois transformed the question of solvability by radicals from a problem concerning fields to a problem about groups. What we do is to change the inquiry into several problems concerning the solvability of certain primitive groups.