Linear differential equations, iterative logarithms and orderings on monomial differential extensions

摘要

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.