Computing automorphisms of abelian number fields

Let be an abelian number field of degree . Most algorithms for computing the lattice of subfields of require the computation of all the conjugates of . This is usually achieved by factoring the minimal polynomial of over . In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of , which is based on -adic techniques. Given and a rational prime which does not divide the discriminant of , the algorithm computes the Frobenius automorphism of in time polynomial in the size of and in the size of . By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of .