Solvability by radicals is in polynomial time

Abstract

Every high school student knows how to express the roots of a quadratic equation in terms of radicals; what is less wellknown 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 quartie 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], hoping to shed some light on the problem, l~variste Galois, the young mathematician who died in a duel at the age of twenty, solved it. In the notes he revised hastily the night before his death, he gave an algorithm which determines when a polynomial has roots expressible in terms of radicals. Yet of this algorithm~ he wrote, '~If now you give me an equation which you have chosen at your pleasure, and if you want to know if it is or is not solvable by radicals, I need do nothing more than to indicate to myself or anyone else the task of doing it. In a word, the calculations are impractical." [Ga].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 ~,ntil the recent Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. © 1983 ACM 0-89791-099-0/83/004/0140 $00.75 breakthrough of Lenstra, Lenstra, and Lovgsz [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. We answer that the basic question of Galois theory -is a given polynomial, f(z), over the rationals solvable by radicals -has a polynomial time solution.Galois transformed the question of sdvability 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. P~lfy has recently shown that the order of a primitive solvable group of degree n is bounded by 24-1/3n c for a constant c -~-3.24399... lPa.] We attempt to construct the Galois group of specified polynomials in polynomial time. Each polynomial is constructed so that its Galois group acts primitively on its roots. If we succeed, we use an algorithm of Sims to determine if the groups in question are solvable. If any one of them is not, the Galois group of f(z) over Q is not solvable, and hence f(x) is n...

Upper Bounds on the Complexity of Some Galois Theory Problems

“…An important breakthrough was achieved by Landau and Miller when they gave a deterministic polynomial time algorithm to check whether a polynomial is solvable by radicals without actually computing the Galois group (see. [6]). We make use of results from [6].…”

“…In [6] it is shown that p(x) ∈ Q(α 1 )[x] and there is a polynomial time deterministic algorithm to find p(x): the algorithm computes each coefficient δ i as a polynomial p i (α 1 ) with rational coefficients. In polynomial time we can compute a primitive element β 1 of Q(δ 0 , δ 1 , .…”

Solving the Closest Vector Problem in 2^n Time -- The Discrete Gaussian Strikes Again!

“…Such algorithms have found applications in factoring polynomials over rationals [3], integer programming [4], [5], [6], cryptanalysis [7], [8], [9], checking the solvability by radicals [10], and solving low-density subset-sum problems [11]. More recently, many powerful cryptographic primitives have been constructed whose security is based on the worst-case hardness of these or related lattice problems [12], [13], [14], [15], [16], [17], [18].…”

On computing subfields. A detailed description of the algorithm

“…We give a detailed description for all parts of the algorithm. 1 There are several other algorithms [1,3,8,9,12,14,15] for calculating subfields. In this article we improve the methods described in [12].…”

“…Three other methods [9,14,15] need factorizations of polynomials over number fields, respectively factorizations of polynomials over the rational integers of much higher degree than the degree of the given field. The method presented in [1] needs hard numerical computations and lattice reduction algorithms.…”