Computing automorphisms of abelian number fields

Abstract

Abstract. Let L = Q(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x) of α over L. 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 α, which is based on p-adic techniques. Given mα(x) and a rational prime p which does not divide the discriminant disc(mα(x)) of mα(x), the algorithm computes the Frobenius automorphism of p in time polynomial in the size of p and in the size of mα(x). By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of α.

An efficient algorithm for the computation of Galois automorphisms

“…A suitable value for D can be a common multiple of the denominators of the entries of the matrix giving an integral basis of K on the powers of α. For B we can use theoretical bounds which depend only of the coefficients of T as in [2], but practical computations show that it is faster to spend time to compute complex approximation for the roots of T and derive a more accurate bound using the following lemma. Let us denote by V the L ∞ -norm when V is a vector, and by M the functional L ∞ -norm when M is a matrix, so that we have the inequality M V ≤ M V , which is simply the supremum of the L 1 -norms of the rows of M .…”

“…In [2], V. Acciaro and J. Klüners give an algorithm for finding Galois automorphisms in the case where the Galois group is Abelian. It can be summarized as follows.…”

“…• the use of LLL algorithms, either real or p-adic (see for example [4, page 174]), • explicit factorization of T (X) in the number field K (see for example [12, page 19]), • the use of unramified p-adic extensions and block system elimination (see [6]), • in the Abelian case, the lift of Frobenius automorphisms (see [2]). …”

“…The above combinatorial techniques are mostly unnecessary in the Abelian case; hence, even though our algorithm can be considered as a modification of the method of [2] which is still valid in the non-Abelian case, its essential ideas are fundamentally different. Even in the Abelian case, for reasonable-sized groups we use a more efficient strategy to find the automorphisms.…”

“…The main advantage of our algorithm is that it is much faster than the LLL or factorization methods, and of comparable speed with the Frobenius lifting methods of [2] in the Abelian case. In [6], one can find a generalization of the Frobenius lifting method to the non-Abelian case.…”