MATHEMATICS OF COMPUTATION Volume 68, Number 226, April 1999, Pages 881–885 S 0025-5718(99)01060-1 SOLVING POLYNOMIALS BY RADICALS WITH ROOTS OF UNITY IN MINIMUM DEPTH GWOBOA HORNG AND MING-DEH HUANG Abstract. Let k be an algebraic number field. Let α be a root of a polynomial f ∈ k[x] which is solvable by radicals. Let L be the splitting field of α over k. Let n be a natural number divisible by the discriminant of the maximal abelian subextension of L, as well as the exponent of G(L/k), the Galois group of L over k. We show that an optimal nested radical with roots of unity for α can be effectively constructed from the derived series of the solvable Galois group of L(ζn) over k(ζn). 1. Introduction It was shown in [8] that whether a polynomial with rational coefficients is solvable by radicals can be decided in polynomial time. Given that a polynomial is solvable by radicals, it is also of interest to construct a nested radical of minimum possible depth for the polynomial. Partial results for this problem can be found in [2, 6, 7, 11]. More recently, a general solution to the problem has been reported in [5]. An interesting relaxation for the problem is to allow roots of unity, in addition to elements of the ground field, to be used as primitives in the construction of nested radicals. No restriction is placed on the roots of unity that can be used for the construction. The goal of this paper is to determine a root of unity for constructing a nested radical of minimum depth for a root of a polynomial which is solvable by radicals. Throughout this paper, k denotes an algebraic number field, ¯ k the algebraic closure of k, μ ∞ the set of all roots of unity, and ζ n = e 2πi/n . Let α be a root of a polynomial f ∈ k[x] that is solvable by radicals. Let L be the splitting field of α over k. Let L ∞ be the splitting field of α over k(μ ∞ ). A near-optimal construction of a nested radical with roots of unity for α is given in [7]. It is also shown in [7] that the minimum depth of a nested radical with roots of unity for α is determined by the length of the derived series of the solvable Galois group of L ∞ over k(μ ∞ ). To effectively construct an optimal nested radical for α, it is desirable to have a similar characterization in terms of a specific root of unity. Let n be a natural number divisible by the discriminant of the maximal abelian subextension of L, as well as the exponent of G(L/k), the Galois group of L over k. We show that the minimum depth of a nested radical with roots of unity for α Received by the editor April 24, 1996 and, in revised form, December 1, 1997. 1991 Mathematics Subject Classification. Primary 11R32; Secondary 11Y16, 12Y05. Key words and phrases. Polynomials, solvable by radicals. The first author was supported in part by NSF Grant CCR 8957317. The second author was supported in part by NSF Grant CCR 9412383. c 1999 American Mathematical Society 881