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The aim of this paper is to exploit the algorithms of paper Experimental Math. 8 (1999) in order to produce a new algebraic method for computing efficiently absolute Lagrange resolvent, a fundamental tool in constructive algebraic Galois theory. This article is composed of two parts.The main idea of the first part is to break up the calculation of absolute resolvent into smaller computations. Since a multi-resolvent is a factor of a resolvent, the whole resolvent may be computed by means of several multi-resolvents.The idea of the second part is that an irreducible polynomial over Z might be reducible over Z /pZ for certain integer p. So the first part can be applied and then, the Chinese remainder theorem allows to lift up the resolvent over Z. [Copyright &y& Elsevier]

Abstract: This paper describes an algorithm which produces all ideals compatible with a given surjective Frobenius near-splitting. [Copyright &y& Elsevier]

Abstract: In numerous applications involving finite fields, we often need high-order elements. Ideally we should be able to obtain a primitive element for any finite field in reasonable time. However, if the prime factorization of the group order is unknown, we do not know how to achieve the goal. We thus turn our attentions to a less ambitious problem: constructing an element of provably high order. In this paper, we survey various algorithms that find an element of high order for general or special finite fields. [Copyright &y& Elsevier]

In this paper, we propose a semi-numerical algorithm for computing absolute factorization of multivariate polynomials. It is based on some properties appearing after a generic change of coordinate. Using numerical computation, Galois group action and rational approximation, this method provides an efficient probabilistic algorithm for medium degrees. Two implementations are presented and compared to other algorithms. [Copyright &y& Elsevier]