Showing papers on "Generic polynomial published in 1983"
TL;DR: For polynomials of the type Xn + aX3 + bX2 + cX + d with absolute Galois group isomorphic to An, covering so, the case n even, 4 ∤ n, for which explicit equations were not known as mentioned in this paper.
8 citations
7 citations
01 Mar 1983
TL;DR: The divide-and-conquer approach answers the question of solvability without revealing the nature of the group in question; the techniques used include methods from computational complexity and approaches from the theory of finite permutation groups.
Abstract: This thesis presents a polynomial time algorithm for the basic question of Galois theory, checking the solvability by radicals of a monic irreducible polynomial over the integers It also presents polynomial time algorithms for factoring polynomials over algebraic number fields, for computing blocks of imprimitivity of roots of a polynomial under the transitive action of number fields (In all of these algorithms it is assumed that the algebraic number field is given by a primitive element which generates it over the rationals, and that the polynomial in question is monic, with coefficients in the integers) We also show how to express a root in radicals in terms of a straight line program in polynomial time The techniques used include methods from computational complexity and approaches from the theory of finite permutation groups The results presented here rely on the recent work of Lenstra, Lenstra, and Lovasz, in which a polynomial time algorithm for factoring polynomials over the integers is presented Many questions remain Our divide-and-conquer approach answers the question of solvability without revealing the nature of the group in question; we do not even learn its order We suggest this as one of the many open problems that remain to be tackled
1 citations