Kant V4

“…We print one nontrivial automorphism to give an idea of the size of the coefficients. This automorphism sends α to 1 165905131392 ( − 1460330046379α + 3045184412874α 3 − 1986742065521α 5 + 506400762490α 7 − 58437366385α 9 + 3179647862α 11 − 78588459α 13 + 701446α 15 ). We can compute the automorphism group of abelian fields of much higher degree, as the following examples, which were given to us by F. Nicolae, show.…”

Section: Examplesmentioning

confidence: 99%

“…The algorithm described in this paper has been implemented using the number theory package KASH [5], developed in Berlin by Prof. M. Pohst and his collaborators.…”

Section: Introductionmentioning

confidence: 99%

Computing automorphisms of abelian number fields

Acciaro

Klüners²

1999

Math. Comp.

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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.

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“…None of them is fundamentally new: many of these ideas have been used elsewhere, e.g. in the computer systems Kant/KASH [14] and Pari/Gp [29]. But, to our knowledge, they do not appear in this form in the literature.…”

Section: Introduction and Notationsmentioning

confidence: 99%