On Computing Galois Groups and Its Application To Solvability by Radicals

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 the Galois group on the roots of the polynomial, and for computing intersections algebraic number fields.