A method for deciding whether the Galois group is abelian
Pilar Fernandez-Ferreiros, M. Ángeles Gómez-Molleda · 2000
We propose a polynomial time algorithm to decide whether the Galois group of an irreducible polynomial ƒ ∈ Q[x] is abelian, and, if so, determine all its elements along with their action on the set of roots of ƒ. This algorithm does not require factorization of polynomials over number fields. Instead we shall use the quadratic Newton—Lifting and the truncated expressions of the roots of ƒ over a p—adic number field Qp, for an appropriate prime p in Z.