Computing automorphisms of abelian number fields期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Computing automorphisms of abelian number fields

Authors:	Vincenzo Acciaro Jü rgen Klü ners.

Affiliation:	Dipartimento di Informatica, Università degli Studi di Bari, via E. Orabona 4, Bari 70125, Italy ; Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany

Abstract:	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 $m_{alpha}(x)$ 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 $m_{alpha}(x)$ and a rational prime which does not divide the discriminant $operatorname{disc} (m_{alpha}(x))$ of $m_{alpha}(x)$ , the algorithm computes the Frobenius automorphism of in time polynomial in the size of and in the size of $m_{alpha}(x)$ . By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of .

Keywords:	Computational number theory abelian number fields automorphisms

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